Subido por Po Chiu

1300 Math Formulas by Golden Art

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```http://fribok.blogspot.com/
1300 Math Formulas
fp_k VVQVNMTTQN
`&ccedil;&eacute;&oacute;&ecirc;&aacute;&Ouml;&Uuml;&iacute; &laquo; OMMQ ^Kp&icirc;&aacute;&ecirc;&aacute;&aring;K ^&auml;&auml; o&aacute;&Ouml;&Uuml;&iacute;&euml; o&Eacute;&euml;&Eacute;&ecirc;&icirc;&Eacute;&Ccedil;K
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Preface
q&Uuml;&aacute;&euml; &Uuml;~&aring;&Ccedil;&Auml;&ccedil;&ccedil;&acirc; &aacute;&euml; ~ &Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&iacute;&Eacute; &Ccedil;&Eacute;&euml;&acirc;&iacute;&ccedil;&eacute; &ecirc;&Eacute;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute; &Ntilde;&ccedil;&ecirc; &euml;&iacute;&igrave;&Ccedil;&Eacute;&aring;&iacute;&euml; ~&aring;&Ccedil; &Eacute;&aring;&Ouml;&aacute;&aring;&Eacute;&Eacute;&ecirc;&euml;K f&iacute; &Uuml;~&euml; &Eacute;&icirc;&Eacute;&ecirc;&oacute;&iacute;&Uuml;&aacute;&aring;&Ouml; &Ntilde;&ecirc;&ccedil;&atilde; &Uuml;&aacute;&Ouml;&Uuml; &euml;&Aring;&Uuml;&ccedil;&ccedil;&auml;
&atilde;~&iacute;&Uuml; &iacute;&ccedil; &atilde;~&iacute;&Uuml; &Ntilde;&ccedil;&ecirc; ~&Ccedil;&icirc;~&aring;&Aring;&Eacute;&Ccedil; &igrave;&aring;&Ccedil;&Eacute;&ecirc;&Ouml;&ecirc;~&Ccedil;&igrave;~&iacute;&Eacute;&euml; &aacute;&aring; &Eacute;&aring;&Ouml;&aacute;&aring;&Eacute;&Eacute;&ecirc;&aacute;&aring;&Ouml;I
&Eacute;&Aring;&ccedil;&aring;&ccedil;&atilde;&aacute;&Aring;&euml;I &eacute;&Uuml;&oacute;&euml;&aacute;&Aring;~&auml; &euml;&Aring;&aacute;&Eacute;&aring;&Aring;&Eacute;&euml;I ~&aring;&Ccedil; &atilde;~&iacute;&Uuml;&Eacute;&atilde;~&iacute;&aacute;&Aring;&euml;K q&Uuml;&Eacute; &Eacute;&Auml;&ccedil;&ccedil;&acirc;
&Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&euml; &Uuml;&igrave;&aring;&Ccedil;&ecirc;&Eacute;&Ccedil;&euml; &ccedil;&Ntilde; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;I &iacute;~&Auml;&auml;&Eacute;&euml;I ~&aring;&Ccedil; &Ntilde;&aacute;&Ouml;&igrave;&ecirc;&Eacute;&euml; &Ntilde;&ecirc;&ccedil;&atilde;
k&igrave;&atilde;&Auml;&Eacute;&ecirc; p&Eacute;&iacute;&euml;I ^&auml;&Ouml;&Eacute;&Auml;&ecirc;~I d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&oacute;I q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&oacute;I j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;
~&aring;&Ccedil; a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;I s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;I ^&aring;~&auml;&oacute;&iacute;&aacute;&Aring; d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&oacute;I `~&auml;&Aring;&igrave;&auml;&igrave;&euml;I
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;I p&Eacute;&ecirc;&aacute;&Eacute;&euml;I ~&aring;&Ccedil; m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; q&Uuml;&Eacute;&ccedil;&ecirc;&oacute;K
q&Uuml;&Eacute; &euml;&iacute;&ecirc;&igrave;&Aring;&iacute;&igrave;&ecirc;&Eacute;&Ccedil; &iacute;~&Auml;&auml;&Eacute; &ccedil;&Ntilde; &Aring;&ccedil;&aring;&iacute;&Eacute;&aring;&iacute;&euml;I &auml;&aacute;&aring;&acirc;&euml;I ~&aring;&Ccedil; &auml;~&oacute;&ccedil;&igrave;&iacute; &atilde;~&acirc;&Eacute;
&Ntilde;&aacute;&aring;&Ccedil;&aacute;&aring;&Ouml; &iacute;&Uuml;&Eacute; &ecirc;&Eacute;&auml;&Eacute;&icirc;~&aring;&iacute; &aacute;&aring;&Ntilde;&ccedil;&ecirc;&atilde;~&iacute;&aacute;&ccedil;&aring; &egrave;&igrave;&aacute;&Aring;&acirc; ~&aring;&Ccedil; &eacute;~&aacute;&aring;&auml;&Eacute;&euml;&euml;I &euml;&ccedil; &aacute;&iacute;
&Aring;~&aring; &Auml;&Eacute; &igrave;&euml;&Eacute;&Ccedil; ~&euml; ~&aring; &Eacute;&icirc;&Eacute;&ecirc;&oacute;&Ccedil;~&oacute; &ccedil;&aring;&auml;&aacute;&aring;&Eacute; &ecirc;&Eacute;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute; &Ouml;&igrave;&aacute;&Ccedil;&Eacute;K
ii
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Contents
1 krj_bo pbqp
NKN
p&Eacute;&iacute; f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&aacute;&Eacute;&euml; 1
NKO
p&Eacute;&iacute;&euml; &ccedil;&Ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; 5
NKP
_~&euml;&aacute;&Aring; f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&aacute;&Eacute;&euml; 7
NKQ
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; 8
2 ^idb_o^
OKN
c~&Aring;&iacute;&ccedil;&ecirc;&aacute;&aring;&Ouml; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 12
OKO
m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 13
OKP
m&ccedil;&iuml;&Eacute;&ecirc;&euml; 14
OKQ
o&ccedil;&ccedil;&iacute;&euml; 15
OKR
i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&euml; 16
OKS
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; 18
OKT
f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&aacute;&Eacute;&euml; 19
OKU
`&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil; f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 22
3 dbljbqov
PKN
o&aacute;&Ouml;&Uuml;&iacute; q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; 24
PKO
f&euml;&ccedil;&euml;&Aring;&Eacute;&auml;&Eacute;&euml; q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; 27
PKP
b&egrave;&igrave;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml; q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; 28
PKQ
p&Aring;~&auml;&Eacute;&aring;&Eacute; q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; 29
PKR
p&egrave;&igrave;~&ecirc;&Eacute; 33
PKS
o&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute; 34
PKT
m~&ecirc;~&auml;&auml;&Eacute;&auml;&ccedil;&Ouml;&ecirc;~&atilde; 35
PKU
o&Uuml;&ccedil;&atilde;&Auml;&igrave;&euml; 36
PKV
q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil; 37
PKNM f&euml;&ccedil;&euml;&Aring;&Eacute;&auml;&Eacute;&euml; q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil; 38
PKNN f&euml;&ccedil;&euml;&Aring;&Eacute;&auml;&Eacute;&euml; q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil; &iuml;&aacute;&iacute;&Uuml; f&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; `&aacute;&ecirc;&Aring;&auml;&Eacute; 40
PKNO q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil; &iuml;&aacute;&iacute;&Uuml; f&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; `&aacute;&ecirc;&Aring;&auml;&Eacute; 41
iii
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PKNP
PKNQ
PKNR
PKNS
PKNT
PKNU
PKNV
PKOM
PKON
PKOO
PKOP
PKOQ
PKOR
PKOS
PKOT
PKOU
PKOV
PKPM
PKPN
PKPO
PKPP
PKPQ
PKPR
PKPS
PKPT
PKPU
PKPV
PKQM
h&aacute;&iacute;&Eacute; 42
`&oacute;&Aring;&auml;&aacute;&Aring; n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml; 43
q~&aring;&Ouml;&Eacute;&aring;&iacute;&aacute;~&auml; n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml; 45
d&Eacute;&aring;&Eacute;&ecirc;~&auml; n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml; 46
o&Eacute;&Ouml;&igrave;&auml;~&ecirc; e&Eacute;&ntilde;~&Ouml;&ccedil;&aring; 47
o&Eacute;&Ouml;&igrave;&auml;~&ecirc; m&ccedil;&auml;&oacute;&Ouml;&ccedil;&aring; 48
`&aacute;&ecirc;&Aring;&auml;&Eacute; 50
p&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; ~ `&aacute;&ecirc;&Aring;&auml;&Eacute; 53
p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; ~ `&aacute;&ecirc;&Aring;&auml;&Eacute; 54
`&igrave;&Auml;&Eacute; 55
o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc; m~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil; 56
m&ecirc;&aacute;&euml;&atilde; 57
o&Eacute;&Ouml;&igrave;&auml;~&ecirc; q&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring; 58
o&Eacute;&Ouml;&igrave;&auml;~&ecirc; m&oacute;&ecirc;~&atilde;&aacute;&Ccedil; 59
c&ecirc;&igrave;&euml;&iacute;&igrave;&atilde; &ccedil;&Ntilde; ~ o&Eacute;&Ouml;&igrave;&auml;~&ecirc; m&oacute;&ecirc;~&atilde;&aacute;&Ccedil; 61
o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc; o&aacute;&Ouml;&Uuml;&iacute; t&Eacute;&Ccedil;&Ouml;&Eacute; 62
m&auml;~&iacute;&ccedil;&aring;&aacute;&Aring; p&ccedil;&auml;&aacute;&Ccedil;&euml; 63
o&aacute;&Ouml;&Uuml;&iacute; `&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc; 66
o&aacute;&Ouml;&Uuml;&iacute; `&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc; &iuml;&aacute;&iacute;&Uuml; ~&aring; l&Auml;&auml;&aacute;&egrave;&igrave;&Eacute; m&auml;~&aring;&Eacute; c~&Aring;&Eacute; 68
o&aacute;&Ouml;&Uuml;&iacute; `&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc; `&ccedil;&aring;&Eacute; 69
c&ecirc;&igrave;&euml;&iacute;&igrave;&atilde; &ccedil;&Ntilde; ~ o&aacute;&Ouml;&Uuml;&iacute; `&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc; `&ccedil;&aring;&Eacute; 70
p&eacute;&Uuml;&Eacute;&ecirc;&Eacute; 72
p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; `~&eacute; 72
p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; p&Eacute;&Aring;&iacute;&ccedil;&ecirc; 73
p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute; 74
p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; t&Eacute;&Ccedil;&Ouml;&Eacute; 75
b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil; 76
`&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc; q&ccedil;&ecirc;&igrave;&euml; 78
4 qofdlkljbqov
QKN
o~&Ccedil;&aacute;~&aring; ~&aring;&Ccedil; a&Eacute;&Ouml;&ecirc;&Eacute;&Eacute; j&Eacute;~&euml;&igrave;&ecirc;&Eacute;&euml; &ccedil;&Ntilde; ^&aring;&Ouml;&auml;&Eacute;&euml; 80
QKO
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;&euml; ~&aring;&Ccedil; d&ecirc;~&eacute;&Uuml;&euml; &ccedil;&Ntilde; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 81
QKP
p&aacute;&Ouml;&aring;&euml; &ccedil;&Ntilde; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 86
QKQ
q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; `&ccedil;&atilde;&atilde;&ccedil;&aring; ^&aring;&Ouml;&auml;&Eacute;&euml; 87
QKR
j&ccedil;&euml;&iacute; f&atilde;&eacute;&ccedil;&ecirc;&iacute;~&aring;&iacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 88
iv
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QKS
QKT
QKU
QKV
QKNM
QKNN
QKNO
QKNP
QKNQ
QKNR
QKNS
QKNT
QKNU
QKNV
QKOM
QKON
o&Eacute;&Ccedil;&igrave;&Aring;&iacute;&aacute;&ccedil;&aring; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 89
m&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;&aacute;&Aring;&aacute;&iacute;&oacute; &ccedil;&Ntilde; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 90
o&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;&euml; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 90
^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring; ~&aring;&Ccedil; p&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&aacute;&ccedil;&aring; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 91
a&ccedil;&igrave;&Auml;&auml;&Eacute; ^&aring;&Ouml;&auml;&Eacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 92
j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&Eacute; ^&aring;&Ouml;&auml;&Eacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 93
e~&auml;&Ntilde; ^&aring;&Ouml;&auml;&Eacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 94
e~&auml;&Ntilde; ^&aring;&Ouml;&auml;&Eacute; q~&aring;&Ouml;&Eacute;&aring;&iacute; f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&aacute;&Eacute;&euml; 94
q&ecirc;~&aring;&euml;&Ntilde;&ccedil;&ecirc;&atilde;&aacute;&aring;&Ouml; &ccedil;&Ntilde; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; b&ntilde;&eacute;&ecirc;&Eacute;&euml;&euml;&aacute;&ccedil;&aring;&euml; &iacute;&ccedil; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; 95
q&ecirc;~&aring;&euml;&Ntilde;&ccedil;&ecirc;&atilde;&aacute;&aring;&Ouml; &ccedil;&Ntilde; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; b&ntilde;&eacute;&ecirc;&Eacute;&euml;&euml;&aacute;&ccedil;&aring;&euml; &iacute;&ccedil; p&igrave;&atilde; 97
m&ccedil;&iuml;&Eacute;&ecirc;&euml; &ccedil;&Ntilde; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 98
d&ecirc;~&eacute;&Uuml;&euml; &ccedil;&Ntilde; f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 99
m&ecirc;&aacute;&aring;&Aring;&aacute;&eacute;~&auml; s~&auml;&igrave;&Eacute;&euml; &ccedil;&Ntilde; f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 102
o&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;&euml; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 103
q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; 106
o&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;&euml; &iacute;&ccedil; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 106
5 j^qof`bp ^ka abqbojfk^kqp
RKN
a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml; 107
RKO
m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml; &ccedil;&Ntilde; a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml; 109
RKP
j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml; 110
RKQ
l&eacute;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml; &iuml;&aacute;&iacute;&Uuml; j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml; 111
RKR
p&oacute;&euml;&iacute;&Eacute;&atilde;&euml; &ccedil;&Ntilde; i&aacute;&aring;&Eacute;~&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; 114
6 sb`qlop
SKN
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; 118
SKO
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; ^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring; 120
SKP
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; p&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&aacute;&ccedil;&aring; 122
SKQ
p&Aring;~&auml;&aacute;&aring;&Ouml; s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; 122
SKR
p&Aring;~&auml;~&ecirc; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; 123
SKS
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; 125
SKT
q&ecirc;&aacute;&eacute;&auml;&Eacute; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; 127
7 ^k^ivqf` dbljbqov
TKN
l&aring;&Eacute; -a&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; p&oacute;&euml;&iacute;&Eacute;&atilde; 130
v
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TKO
TKP
TKQ
TKR
TKS
TKT
TKU
TKV
TKNM
TKNN
TKNO
q&iuml;&ccedil; -a&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; p&oacute;&euml;&iacute;&Eacute;&atilde; 131
p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute; &aacute;&aring; m&auml;~&aring;&Eacute; 139
`&aacute;&ecirc;&Aring;&auml;&Eacute; 149
b&auml;&auml;&aacute;&eacute;&euml;&Eacute; 152
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~ 154
m~&ecirc;~&Auml;&ccedil;&auml;~ 158
q&Uuml;&ecirc;&Eacute;&Eacute; -a&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; p&oacute;&euml;&iacute;&Eacute;&atilde; 161
m&auml;~&aring;&Eacute; 165
p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute; &aacute;&aring; p&eacute;~&Aring;&Eacute; 175
n&igrave;~&Ccedil;&ecirc;&aacute;&Aring; p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;&euml; 180
p&eacute;&Uuml;&Eacute;&ecirc;&Eacute; 189
8 afccbobkqf^i `^i`rirp
UKN
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; ~&aring;&Ccedil; q&Uuml;&Eacute;&aacute;&ecirc; d&ecirc;~&eacute;&Uuml;&euml; 191
UKO
i&aacute;&atilde;&aacute;&iacute;&euml; &ccedil;&Ntilde; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 208
UKP
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; ~&aring;&Ccedil; m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; 209
UKQ
q~&Auml;&auml;&Eacute; &ccedil;&Ntilde; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml; 211
UKR
e&aacute;&Ouml;&Uuml;&Eacute;&ecirc; l&ecirc;&Ccedil;&Eacute;&ecirc; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml; 215
UKS
^&eacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; 217
UKT
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; 221
UKU
j&igrave;&auml;&iacute;&aacute;&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 222
UKV
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; l&eacute;&Eacute;&ecirc;~&iacute;&ccedil;&ecirc;&euml; 225
9 fkqbdo^i `^i`rirp
VKN
f&aring;&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; 227
VKO
f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&Ntilde; o~&iacute;&aacute;&ccedil;&aring;~&auml; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 228
VKP
f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&Ntilde; f&ecirc;&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 231
VKQ
f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&Ntilde; q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 237
VKR
f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&Ntilde; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 241
VKS
f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&Ntilde; b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml; ~&aring;&Ccedil; i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 242
VKT
o&Eacute;&Ccedil;&igrave;&Aring;&iacute;&aacute;&ccedil;&aring; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 243
VKU
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; 247
VKV
f&atilde;&eacute;&ecirc;&ccedil;&eacute;&Eacute;&ecirc; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; 253
VKNM a&ccedil;&igrave;&Auml;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; 257
VKNN q&ecirc;&aacute;&eacute;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; 269
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VKNO
VKNP
i&aacute;&aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; 275
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; 285
10 afccbobkqf^i bnr^qflkp
NMKN c&aacute;&ecirc;&euml;&iacute; l&ecirc;&Ccedil;&Eacute;&ecirc; l&ecirc;&Ccedil;&aacute;&aring;~&ecirc;&oacute; a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; 295
NMKO p&Eacute;&Aring;&ccedil;&aring;&Ccedil; l&ecirc;&Ccedil;&Eacute;&ecirc; l&ecirc;&Ccedil;&aacute;&aring;~&ecirc;&oacute; a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; 298
NMKP p&ccedil;&atilde;&Eacute; m~&ecirc;&iacute;&aacute;~&auml; a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; 302
11 pbofbp
NNKN ^&ecirc;&aacute;&iacute;&Uuml;&atilde;&Eacute;&iacute;&aacute;&Aring; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 304
NNKO d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 305
NNKP p&ccedil;&atilde;&Eacute; c&aacute;&aring;&aacute;&iacute;&Eacute; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 305
NNKQ f&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 307
NNKR m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml; &ccedil;&Ntilde; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 307
NNKS `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; q&Eacute;&euml;&iacute;&euml; 308
NNKT ^&auml;&iacute;&Eacute;&ecirc;&aring;~&iacute;&aacute;&aring;&Ouml; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 310
NNKU m&ccedil;&iuml;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 311
NNKV a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&iacute;&aacute;&ccedil;&aring; ~&aring;&Ccedil; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; m&ccedil;&iuml;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 312
NNKNM q~&oacute;&auml;&ccedil;&ecirc; ~&aring;&Ccedil; j~&Aring;&auml;~&igrave;&ecirc;&aacute;&aring; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 313
NNKNN m&ccedil;&iuml;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml; b&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring;&euml; &Ntilde;&ccedil;&ecirc; p&ccedil;&atilde;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; 314
NNKNO _&aacute;&aring;&ccedil;&atilde;&aacute;~&auml; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 316
NNKNP c&ccedil;&igrave;&ecirc;&aacute;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml; 316
12 mol_^_fifqv
NOKN m&Eacute;&ecirc;&atilde;&igrave;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml; ~&aring;&Ccedil; `&ccedil;&atilde;&Auml;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;&euml; 318
NOKO m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; 319
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Chapter 1
Number Sets
1.1 Set Identities
p&Eacute;&iacute;&euml;W ^I _I `
r&aring;&aacute;&icirc;&Eacute;&ecirc;&euml;~&auml; &euml;&Eacute;&iacute;W f
`&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; W ^′
m&ecirc;&ccedil;&eacute;&Eacute;&ecirc; &euml;&igrave;&Auml;&euml;&Eacute;&iacute;W ^ ⊂ _
b&atilde;&eacute;&iacute;&oacute; &euml;&Eacute;&iacute;W ∅
r&aring;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &euml;&Eacute;&iacute;&euml;W ^ ∪ _
f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &euml;&Eacute;&iacute;&euml;W ^ ∩ _
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; &euml;&Eacute;&iacute;&euml;W ^ y _
1.
^⊂f
2.
^⊂^
3.
^ = _ &aacute;&Ntilde; ^ ⊂ _ ~&aring;&Ccedil; _ ⊂ ^ .
4.
b&atilde;&eacute;&iacute;&oacute; p&Eacute;&iacute;
∅⊂^
5.
r&aring;&aacute;&ccedil;&aring; &ccedil;&Ntilde; p&Eacute;&iacute;&euml;
` = ^ ∪ _ = {&ntilde; &ouml; &ntilde; ∈ ^ &ccedil;&ecirc; &ntilde; ∈ _}
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CHAPTER 1. NUMBER SETS
Figure 1.
6.
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;
^∪_ = _∪^
7.
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;
^ ∪ (_ ∪ ` ) = (^ ∪ _ ) ∪ `
8.
f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; p&Eacute;&iacute;&euml;
` = ^ ∪ _ = {&ntilde; &ouml; &ntilde; ∈ ^ ~&aring;&Ccedil; &ntilde; ∈ _}
Figure 2.
9.
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;
^∩_ = _∩^
10.
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;
^ ∩ (_ ∩ ` ) = (^ ∩ _ ) ∩ `
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CHAPTER 1. NUMBER SETS
11.
a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;
^ ∪ (_ ∩ ` ) = (^ ∪ _ ) ∩ (^ ∪ ` ) I
^ ∩ (_ ∪ ` ) = (^ ∩ _ ) ∪ (^ ∩ ` ) K
12.
f&Ccedil;&Eacute;&atilde;&eacute;&ccedil;&iacute;&Eacute;&aring;&Aring;&oacute;
^∩^ = ^I
^∪^= ^
13.
a&ccedil;&atilde;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;
^∩∅ = ∅I
^∪f= f
14.
f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&oacute;
^ ∪∅ = ^ I
^∩f= ^
15.
`&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;
^′ = {&ntilde; ∈ f &ouml; &ntilde; ∉ ^}
16.
`&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; ~&aring;&Ccedil; r&aring;&aacute;&ccedil;&aring;
^ ∪ ^′ = f I
^ ∩ ^′ = ∅
17.
a&Eacute; j&ccedil;&ecirc;&Ouml;~&aring;∞&euml; i~&iuml;&euml;
(^ ∪ _ )′ = ^′ ∩ _′ I
(^ ∩ _ )′ = ^′ ∪ _′
18.
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; p&Eacute;&iacute;&euml;
` = _ y ^ = {&ntilde; &ouml; &ntilde; ∈ _ ~&aring;&Ccedil; &ntilde; ∉ ^}
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CHAPTER 1. NUMBER SETS
Figure 3.
19.
_ y ^ = _ y (^ ∩ _ )
20.
_ y ^ = _ ∩ ^′
21.
^y^=∅
22.
^ y _ = ^ &aacute;&Ntilde; ^ ∩ _ = ∅ .
Figure 4.
23.
(^ y _) ∩ ` = (^ ∩ `) y (_ ∩ `)
24.
^′ = f y ^
25.
`~&ecirc;&iacute;&Eacute;&euml;&aacute;~&aring; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;
` = ^ &times; _ = {(&ntilde; I &oacute; ) &ouml; &ntilde; ∈ ^ ~&aring;&Ccedil; &oacute; ∈ _}
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CHAPTER 1. NUMBER SETS
1.2 Sets of Numbers
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W k
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W kM
f&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;W w
m&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;W w +
k&Eacute;&Ouml;~&iacute;&aacute;&icirc;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;W w −
o~&iacute;&aacute;&ccedil;&aring;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W n
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W o
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W `
26.
k~&iacute;&igrave;&ecirc;~&auml; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
`&ccedil;&igrave;&aring;&iacute;&aacute;&aring;&Ouml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W k = {NI OI PI K} K
27.
t&Uuml;&ccedil;&auml;&Eacute; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
`&ccedil;&igrave;&aring;&iacute;&aacute;&aring;&Ouml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; ~&aring;&Ccedil; &ograve;&Eacute;&ecirc;&ccedil;W k M = {MI NI OI PI K} K
28.
f&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; ~&aring;&Ccedil; &iacute;&Uuml;&Eacute;&aacute;&ecirc; &ccedil;&eacute;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute;&euml; ~&aring;&Ccedil; &ograve;&Eacute;&ecirc;&ccedil;W
w + = k = {NI OI PI K}I
w − = {KI − PI − OI − N} I
w = w − ∪ {M} ∪ w + = {KI − PI − OI − NI MI NI OI PI K} K
29.
o~&iacute;&aacute;&ccedil;&aring;~&auml; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
o&Eacute;&eacute;&Eacute;~&iacute;&aacute;&aring;&Ouml; &ccedil;&ecirc; &iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&iacute;&aacute;&aring;&Ouml; &Ccedil;&Eacute;&Aring;&aacute;&atilde;~&auml;&euml;W
~


n = &ntilde; &ouml; &ntilde; = ~&aring;&Ccedil; ~ ∈ w ~&aring;&Ccedil; &Auml; ∈ w ~&aring;&Ccedil; &Auml; ≠ M K
&Auml;


30.
f&ecirc;&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
k&ccedil;&aring;&ecirc;&Eacute;&eacute;&Eacute;~&iacute;&aacute;&aring;&Ouml; ~&aring;&Ccedil; &aring;&ccedil;&aring;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&iacute;&aacute;&aring;&Ouml; &Ccedil;&Eacute;&Aring;&aacute;&atilde;~&auml;&euml;K
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CHAPTER 1. NUMBER SETS
31.
o&Eacute;~&auml; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
r&aring;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml; ~&aring;&Ccedil; &aacute;&ecirc;&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W oK
32.
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
` = {&ntilde; + &aacute;&oacute; &ouml; &ntilde; ∈ o ~&aring;&Ccedil; &oacute; ∈ o}I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &aacute; &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; &igrave;&aring;&aacute;&iacute;K
33.
k⊂w⊂n⊂o⊂`
Figure 5.
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CHAPTER 1. NUMBER SETS
1.3 Basic Identities
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ~I &Auml;I &Aring;
34.
^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&icirc;&Eacute; f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&oacute;
~+M=~
35.
^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&icirc;&Eacute; f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;
~ + (− ~ ) = M
36.
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute; &ccedil;&Ntilde; ^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;
~ +&Auml;= &Auml;+~
37.
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute; &ccedil;&Ntilde; ^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;
(~ + &Auml;) + &Aring; = ~ + (&Auml; + &Aring; )
38.
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; p&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&aacute;&ccedil;&aring;
~ − &Auml; = ~ + (− &Auml;)
39.
j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&icirc;&Eacute; f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&oacute;
~ ⋅N = ~
40.
j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&icirc;&Eacute; f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;
N
~ ⋅ =NI ~ ≠ M
~
41.
j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring; q&aacute;&atilde;&Eacute;&euml; M
~ ⋅M = M
42.
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute; &ccedil;&Ntilde; j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;
~ ⋅&Auml; = &Auml;⋅~
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CHAPTER 1. NUMBER SETS
43.
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute; &ccedil;&Ntilde; j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;
(~ ⋅ &Auml;)⋅ &Aring; = ~ ⋅ (&Auml; ⋅ &Aring; )
44.
a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&Eacute; i~&iuml;
~ (&Auml; + &Aring; ) = ~&Auml; + ~&Aring;
45.
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; a&aacute;&icirc;&aacute;&euml;&aacute;&ccedil;&aring;
~
N
= ~⋅
&Auml;
&Auml;
1.4 Complex Numbers
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; &igrave;&aring;&aacute;&iacute;W &aacute;
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &ograve;
o&Eacute;~&auml; &eacute;~&ecirc;&iacute;W ~I &Aring;
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; &eacute;~&ecirc;&iacute;W &Auml;&aacute;I &Ccedil;&aacute;
j&ccedil;&Ccedil;&igrave;&auml;&igrave;&euml; &ccedil;&Ntilde; ~ &Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &ecirc;I &ecirc;N I &ecirc;O
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; ~ &Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W ϕ I ϕN I ϕO
46.
&aacute;N = &aacute;
&aacute; O = −N
&aacute; P = −&aacute;
&aacute;Q = N
&aacute;R = &aacute;
&aacute; S = −N
&aacute; T = −&aacute;
&aacute;U = N
47.
&ograve; = ~ + &Auml;&aacute;
48.
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; m&auml;~&aring;&Eacute;
&aacute; Q &aring; +N = &aacute;
&aacute; Q &aring;+ O = −N
&aacute; Q &aring; + P = −&aacute;
&aacute; Q&aring; = N
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CHAPTER 1. NUMBER SETS
Figure 6.
49.
(~ + &Auml;&aacute; ) + (&Aring; + &Ccedil;&aacute; ) = (~ + &Aring; ) + (&Auml; + &Ccedil; )&aacute;
50.
(~ + &Auml;&aacute; ) − (&Aring; + &Ccedil;&aacute; ) = (~ − &Aring; ) + (&Auml; − &Ccedil;)&aacute;
51.
(~ + &Auml;&aacute; )(&Aring; + &Ccedil;&aacute; ) = (~&Aring; − &Auml;&Ccedil; ) + (~&Ccedil; + &Auml;&Aring; )&aacute;
52.
~ + &Auml;&aacute; ~&Aring; + &Auml;&Ccedil; &Auml;&Aring; − ~&Ccedil;
=
+
⋅&aacute;
&Aring; + &Ccedil;&aacute; &Aring; O + &Ccedil; O &Aring; O + &Ccedil; O
53.
`&ccedil;&aring;&agrave;&igrave;&Ouml;~&iacute;&Eacute; `&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
|||||||
~ + &Auml;&aacute; = ~ − &Auml;&aacute;
54.
~ = &ecirc; &Aring;&ccedil;&euml; ϕ I &Auml; = &ecirc; &euml;&aacute;&aring; ϕ
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CHAPTER 1. NUMBER SETS
Figure 7.
55.
m&ccedil;&auml;~&ecirc; m&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; `&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
~ + &Auml;&aacute; = &ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ)
56.
j&ccedil;&Ccedil;&igrave;&auml;&igrave;&euml; ~&aring;&Ccedil; ^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; ~ `&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;
f&Ntilde; ~ + &Auml;&aacute; &aacute;&euml; ~ &Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;I &iacute;&Uuml;&Eacute;&aring;
&ecirc; = ~ O + &Auml;O E&atilde;&ccedil;&Ccedil;&igrave;&auml;&igrave;&euml;FI
&Auml;
ϕ = ~&ecirc;&Aring;&iacute;~&aring; E~&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;FK
~
57.
m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; &aacute;&aring; m&ccedil;&auml;~&ecirc; o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;
&ograve; N ⋅ &ograve; O = &ecirc;N (&Aring;&ccedil;&euml; ϕN + &aacute; &euml;&aacute;&aring; ϕN ) ⋅ &ecirc;O (&Aring;&ccedil;&euml; ϕO + &aacute; &euml;&aacute;&aring; ϕO )
= &ecirc;N&ecirc;O [&Aring;&ccedil;&euml;(ϕN + ϕO ) + &aacute; &euml;&aacute;&aring;(ϕN + ϕO )]
58.
`&ccedil;&aring;&agrave;&igrave;&Ouml;~&iacute;&Eacute; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; &aacute;&aring; m&ccedil;&auml;~&ecirc; o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;
|||||||||||||||||||||
&ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ) = &ecirc;[&Aring;&ccedil;&euml;(− ϕ) + &aacute; &euml;&aacute;&aring;(− ϕ)]
59.
f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; &ccedil;&Ntilde; ~ `&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc; &aacute;&aring; m&ccedil;&auml;~&ecirc; o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;
N
N
= [&Aring;&ccedil;&euml;(− ϕ) + &aacute; &euml;&aacute;&aring;(− ϕ)]
&ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ) &ecirc;
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60.
n&igrave;&ccedil;&iacute;&aacute;&Eacute;&aring;&iacute; &aacute;&aring; m&ccedil;&auml;~&ecirc; o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;
&ograve; N &ecirc;N (&Aring;&ccedil;&euml; ϕN + &aacute; &euml;&aacute;&aring; ϕN ) &ecirc;N
=
= [&Aring;&ccedil;&euml;(ϕN − ϕO ) + &aacute; &euml;&aacute;&aring;(ϕN − ϕO )]
&ograve; O &ecirc;O (&Aring;&ccedil;&euml; ϕO + &aacute; &euml;&aacute;&aring; ϕO ) &ecirc;O
61.
m&ccedil;&iuml;&Eacute;&ecirc; &ccedil;&Ntilde; ~ `&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;
&aring;
&ograve; &aring; = [&ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ)] = &ecirc; &aring; [&Aring;&ccedil;&euml;(&aring;ϕ) + &aacute; &euml;&aacute;&aring;(&aring;ϕ)]
62.
c&ccedil;&ecirc;&atilde;&igrave;&auml;~ &plusmn;a&Eacute; j&ccedil;&aacute;&icirc;&ecirc;&Eacute;≤
(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ)&aring; = &Aring;&ccedil;&euml;(&aring;ϕ) + &aacute; &euml;&aacute;&aring;(&aring;ϕ)
63.
k&iacute;&Uuml; o&ccedil;&ccedil;&iacute; &ccedil;&Ntilde; ~ `&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; k&igrave;&atilde;&Auml;&Eacute;&ecirc;
ϕ + Oπ&acirc;
ϕ + Oπ&acirc; 

&aring;
&ograve; = &aring; &ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ) = &aring; &ecirc;  &Aring;&ccedil;&euml;
+ &aacute; &euml;&aacute;&aring;
I
&aring;
&aring; 

&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&acirc; = MI NI OI KI &aring; − N K
64.
b&igrave;&auml;&Eacute;&ecirc;∞&euml; c&ccedil;&ecirc;&atilde;&igrave;&auml;~
&Eacute; &aacute;&ntilde; = &Aring;&ccedil;&euml; &ntilde; + &aacute; &euml;&aacute;&aring; &ntilde;
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Chapter 2
Algebra
2.1 Factoring Formulas
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ~I &Auml;I &Aring;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
65.
~ O − &Auml;O = (~ + &Auml;)(~ − &Auml;)
66.
~ P − &Auml;P = (~ − &Auml;)(~ O + ~&Auml; + &Auml;O )
67.
~ P + &Auml;P = (~ + &Auml;)(~ O − ~&Auml; + &Auml;O )
68.
~ Q − &Auml;Q = (~ O − &Auml;O )(~ O + &Auml;O ) = (~ − &Auml;)(~ + &Auml;)(~ O + &Auml;O )
69.
~ R − &Auml;R = (~ − &Auml;)(~ Q + ~ P &Auml; + ~ O &Auml;O + ~&Auml;P + &Auml;Q )
70.
~ R + &Auml;R = (~ + &Auml;)(~ Q − ~ P &Auml; + ~ O &Auml;O − ~&Auml;P + &Auml;Q )
71.
f&Ntilde; &aring; &aacute;&euml; &ccedil;&Ccedil;&Ccedil;I &iacute;&Uuml;&Eacute;&aring;
~ &aring; + &Auml;&aring; = (~ + &Auml;)(~ &aring;−N − ~ &aring; −O &Auml; + ~ &aring; −P &Auml;O − K − ~&Auml;&aring; −O + &Auml;&aring; −N ) K
72.
f&Ntilde; &aring; &aacute;&euml; &Eacute;&icirc;&Eacute;&aring;I &iacute;&Uuml;&Eacute;&aring;
~ &aring; − &Auml;&aring; = (~ − &Auml;)(~ &aring; −N + ~ &aring; −O &Auml; + ~ &aring; −P &Auml;O + K + ~&Auml;&aring;−O + &Auml;&aring; −N ) I
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CHAPTER 2. ALGEBRA
~ &aring; + &Auml;&aring; = (~ + &Auml;)(~ &aring;−N − ~ &aring; −O &Auml; + ~ &aring; −P &Auml;O − K + ~&Auml;&aring;−O − &Auml;&aring; −N ) K
2.2 Product Formulas
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ~I &Auml;I &Aring;
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &aring;I &acirc;
73.
(~ − &Auml;)O = ~ O − O~&Auml; + &Auml;O
74.
(~ + &Auml;)O = ~ O + O~&Auml; + &Auml;O
75.
(~ − &Auml;)P = ~ P − P~ O &Auml; + P~&Auml;O − &Auml;P
76.
(~ + &Auml;)P = ~ P + P~ O&Auml; + P~&Auml;O + &Auml;P
77.
(~ − &Auml;)Q = ~ Q − Q~ P &Auml; + S~ O &Auml;O − Q~&Auml;P + &Auml;Q
78.
(~ + &Auml;)Q = ~ Q + Q~ P &Auml; + S~ O&Auml;O + Q~&Auml;P + &Auml;Q
79.
80.
81.
_&aacute;&aring;&ccedil;&atilde;&aacute;~&auml; c&ccedil;&ecirc;&atilde;&igrave;&auml;~
(~ + &Auml;)&aring; = &aring;` M~ &aring; + &aring;`N~ &aring;−N&Auml; + &aring;` O~ &aring;−O&Auml;O + K + &aring;` &aring;−N~&Auml;&aring;−N + &aring;` &aring; &Auml;&aring; I
&aring;&gt;
~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Auml;&aacute;&aring;&ccedil;&atilde;&aacute;~&auml; &Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;K
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &aring; ` &acirc; =
&acirc;&gt; (&aring; − &acirc; )&gt;
(~ + &Auml; + &Aring; )O = ~ O + &Auml;O + &Aring; O + O~&Auml; + O~&Aring; + O&Auml;&Aring;
(~ + &Auml; + &Aring; + K + &igrave; + &icirc; )O = ~ O + &Auml;O + &Aring; O + K + &igrave; O + &icirc; O +
+ O(~&Auml; + ~&Aring; + K + ~&igrave; + ~&icirc; + &Auml;&Aring; + K + &Auml;&igrave; + &Auml;&icirc; + K + &igrave;&icirc; )
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CHAPTER 2. ALGEBRA
2.3 Powers
_~&euml;&Eacute;&euml; E&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &ecirc;&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;FW ~I &Auml;
m&ccedil;&iuml;&Eacute;&ecirc;&euml; E&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;FW &aring;I &atilde;
82.
~ &atilde; ~ &aring; = ~ &atilde;+&aring;
83.
~&atilde;
= ~ &atilde; −&aring;
&aring;
~
84.
(~&Auml;)&atilde; = ~ &atilde; &Auml;&atilde;
85.
~&atilde;
~
  = &atilde;
&Auml;
 &Auml;
86.
(~ )
87.
~M = N I ~ ≠ M
88.
~N = N
89.
~ −&atilde; =
&atilde;
90.
&atilde; &aring;
= ~ &atilde;&aring;
N
~&atilde;
&atilde;
&aring;
~ = &aring; ~&atilde;
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CHAPTER 2. ALGEBRA
2.4 Roots
_~&euml;&Eacute;&euml;W ~I &Auml;
m&ccedil;&iuml;&Eacute;&ecirc;&euml; E&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;FW &aring;I &atilde;
~ I &Auml; ≥ M &Ntilde;&ccedil;&ecirc; &Eacute;&icirc;&Eacute;&aring; &ecirc;&ccedil;&ccedil;&iacute;&euml; E &aring; = O&acirc; I &acirc; ∈ k F
91.
&aring;
~&Auml; = &aring; ~ &aring; &Auml;
92.
&aring;
~ &atilde; &Auml; = &aring;&atilde; ~ &atilde; &Auml;&aring;
93.
&aring;
~ &aring;~
=
I &Auml;≠M
&Auml; &aring;&Auml;
94.
~ &aring;&atilde; ~ &atilde; &aring;&atilde; ~ &atilde;
=
=
I &Auml;≠MK
&atilde;
&Auml;&aring;
&Auml; &aring;&atilde; &Auml;&aring;
&aring;
95.
(~ )
96.
( ~)
&aring;
&atilde;
&aring;
&aring;
&eacute;
= &aring; ~ &atilde;&eacute;
=~
&aring;&eacute;
97.
&aring;
~&atilde; =
98.
&aring;
~ =~
99.
&atilde; &aring;
100.
&atilde;
&atilde;
&aring;
~ = &atilde;&aring; ~
( ~)
&aring;
~ &atilde;&eacute;
&atilde;
= &aring; ~&atilde;
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CHAPTER 2. ALGEBRA
101.
N &aring; ~ &aring; −N
=
I ~ ≠ MK
&aring;
~
~
~ + ~O − &Auml;
~ − ~O − &Auml;
&plusmn;
O
O
102.
~&plusmn; &Auml; =
103.
N
~m &Auml;
=
~−&Auml;
~&plusmn; &Auml;
2.5 Logarithms
m&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &ecirc;&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &ntilde;I &oacute;I ~I &Aring;I &acirc;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
104. a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;
&oacute; = &auml;&ccedil;&Ouml; ~ &ntilde; &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde; &ntilde; = ~ &oacute; I ~ &gt; M I ~ ≠ N K
105. &auml;&ccedil;&Ouml; ~ N = M
106. &auml;&ccedil;&Ouml; ~ ~ = N
− ∞ &aacute;&Ntilde; ~ &gt; N
107. &auml;&ccedil;&Ouml; ~ M = 
+ ∞ &aacute;&Ntilde; ~ &lt; N
108. &auml;&ccedil;&Ouml; ~ (&ntilde;&oacute; ) = &auml;&ccedil;&Ouml; ~ &ntilde; + &auml;&ccedil;&Ouml; ~ &oacute;
109. &auml;&ccedil;&Ouml; ~
&ntilde;
= &auml;&ccedil;&Ouml; ~ &ntilde; − &auml;&ccedil;&Ouml; ~ &oacute;
&oacute;
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CHAPTER 2. ALGEBRA
110. &auml;&ccedil;&Ouml; ~ (&ntilde; &aring; ) = &aring; &auml;&ccedil;&Ouml; ~ &ntilde;
111. &auml;&ccedil;&Ouml; ~ &aring; &ntilde; =
N
&auml;&ccedil;&Ouml; ~ &ntilde;
&aring;
112. &auml;&ccedil;&Ouml; ~ &ntilde; =
&auml;&ccedil;&Ouml; &Aring; &ntilde;
= &auml;&ccedil;&Ouml; &Aring; &ntilde; ⋅ &auml;&ccedil;&Ouml; ~ &Aring; I &Aring; &gt; M I &Aring; ≠ N K
&auml;&ccedil;&Ouml; &Aring; ~
113. &auml;&ccedil;&Ouml; ~ &Aring; =
N
&auml;&ccedil;&Ouml; &Aring; ~
114. &ntilde; = ~ &auml;&ccedil;&Ouml; ~ &ntilde;
115. i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde; &iacute;&ccedil; _~&euml;&Eacute; NM
&auml;&ccedil;&Ouml; NM &ntilde; = &auml;&ccedil;&Ouml; &ntilde;
116. k~&iacute;&igrave;&ecirc;~&auml; i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;
&auml;&ccedil;&Ouml; &Eacute; &ntilde; = &auml;&aring; &ntilde; I
&acirc;
 N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Eacute; = &auml;&aacute;&atilde; N +  = OKTNUOUNUOUK
&acirc; →∞
 &acirc;
117. &auml;&ccedil;&Ouml; &ntilde; =
118. &auml;&aring; &ntilde; =
N
&auml;&aring; &ntilde; = MKQPQOVQ &auml;&aring; &ntilde;
&auml;&aring; NM
N
&auml;&ccedil;&Ouml; &ntilde; = OKPMORUR &auml;&ccedil;&Ouml; &ntilde;
&auml;&ccedil;&Ouml; &Eacute;
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CHAPTER 2. ALGEBRA
2.6 Equations
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ~I &Auml;I &Aring;I &eacute;I &egrave;I &igrave;I &icirc;
p&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;W &ntilde; N I &ntilde; O I &oacute; N I &oacute; O I &oacute; P
119. i&aacute;&aring;&Eacute;~&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &aacute;&aring; l&aring;&Eacute; s~&ecirc;&aacute;~&Auml;&auml;&Eacute;
&Auml;
~&ntilde; + &Auml; = M I &ntilde; = − K
~
120. n&igrave;~&Ccedil;&ecirc;~&iacute;&aacute;&Aring; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
− &Auml; &plusmn; &Auml;O − Q~&Aring;
~&ntilde; + &Auml;&ntilde; + &Aring; = M I &ntilde; NI O =
K
O~
O
121. a&aacute;&euml;&Aring;&ecirc;&aacute;&atilde;&aacute;&aring;~&aring;&iacute;
a = &Auml;O − Q~&Aring;
122. s&aacute;&Eacute;&iacute;&Eacute;∞&euml; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;
f&Ntilde; &ntilde; O + &eacute;&ntilde; + &egrave; = M I &iacute;&Uuml;&Eacute;&aring;
&ntilde; N + &ntilde; O = −&eacute;
K

&ntilde;
&ntilde;
=
&egrave;
N
O

&Auml;
123. ~&ntilde; O + &Auml;&ntilde; = M I &ntilde; N = M I &ntilde; O = − K
~
124. ~&ntilde; O + &Aring; = M I &ntilde; NI O = &plusmn; −
&Aring;
K
~
125. `&igrave;&Auml;&aacute;&Aring; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;K `~&ecirc;&Ccedil;~&aring;&ccedil;∞&euml; c&ccedil;&ecirc;&atilde;&igrave;&auml;~K
&oacute; P + &eacute;&oacute; + &egrave; = M I
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CHAPTER 2. ALGEBRA
&oacute; N = &igrave; + &icirc; I &oacute; OI P = −
N
(&igrave; + &icirc; ) &plusmn; P (&igrave; + &icirc; ) &aacute; I
O
O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
O
&igrave;=P −
O
O
O
&egrave;
&egrave;
&egrave;  &eacute;
 &egrave;  &eacute;
+   +  I &icirc; =P − −   +  K
O
O
 O P
 O P
2.7 Inequalities
s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &ntilde;I &oacute;I &ograve;
~ I &Auml;I &Aring;I &Ccedil;
I &atilde;I &aring;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W 
~N I ~ O I ~ P I KI ~ &aring;
a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;W aI a&ntilde; I a&oacute; I a&ograve;
126. f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&aacute;&Eacute;&euml;I f&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; k&ccedil;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml; ~&aring;&Ccedil; d&ecirc;~&eacute;&Uuml;&euml;
f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;
~≤&ntilde;≤&Auml;
f&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; k&ccedil;&iacute;~&iacute;&aacute;&ccedil;&aring;
[~I &Auml;]
~&lt;&ntilde;≤&Auml;
(~I &Auml;]
~≤&ntilde;&lt;&Auml;
[~I &Auml;)
~&lt;&ntilde;&lt;&Auml;
(~I &Auml;)
−∞&lt; &ntilde; ≤&Auml;I
&ntilde;≤&Auml;
−∞&lt; &ntilde; &lt;&Auml;I
&ntilde;&lt;&Auml;
~≤&ntilde;&lt;∞I
&ntilde;≥~
~&lt;&ntilde;&lt;∞I
&ntilde; &gt;~
(− ∞I &Auml;]
(− ∞I &Auml;)
[~I ∞ )
(~I ∞ )
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d&ecirc;~&eacute;&Uuml;
CHAPTER 2. ALGEBRA
127. f&Ntilde; ~ &gt; &Auml; I &iacute;&Uuml;&Eacute;&aring; &Auml; &lt; ~ K
128. f&Ntilde; ~ &gt; &Auml; I &iacute;&Uuml;&Eacute;&aring; ~ − &Auml; &gt; M &ccedil;&ecirc; &Auml; − ~ &lt; M K
129. f&Ntilde; ~ &gt; &Auml; I &iacute;&Uuml;&Eacute;&aring; ~ + &Aring; &gt; &Auml; + &Aring; K
130. f&Ntilde; ~ &gt; &Auml; I &iacute;&Uuml;&Eacute;&aring; ~ − &Aring; &gt; &Auml; − &Aring; K
131. f&Ntilde; ~ &gt; &Auml; ~&aring;&Ccedil; &Aring; &gt; &Ccedil; I &iacute;&Uuml;&Eacute;&aring; ~ + &Aring; &gt; &Auml; + &Ccedil; K
132. f&Ntilde; ~ &gt; &Auml; ~&aring;&Ccedil; &Aring; &gt; &Ccedil; I &iacute;&Uuml;&Eacute;&aring; ~ − &Ccedil; &gt; &Auml; − &Aring; K
133. f&Ntilde; ~ &gt; &Auml; ~&aring;&Ccedil; &atilde; &gt; M I &iacute;&Uuml;&Eacute;&aring; &atilde;~ &gt; &atilde;&Auml; K
134. f&Ntilde; ~ &gt; &Auml; ~&aring;&Ccedil; &atilde; &gt; M I &iacute;&Uuml;&Eacute;&aring;
~ &Auml;
&gt; K
&atilde; &atilde;
135. f&Ntilde; ~ &gt; &Auml; ~&aring;&Ccedil; &atilde; &lt; M I &iacute;&Uuml;&Eacute;&aring; &atilde;~ &lt; &atilde;&Auml; K
136. f&Ntilde; ~ &gt; &Auml; ~&aring;&Ccedil; &atilde; &lt; M I &iacute;&Uuml;&Eacute;&aring;
~ &Auml;
&lt; K
&atilde; &atilde;
137. f&Ntilde; M &lt; ~ &lt; &Auml; ~&aring;&Ccedil; &aring; &gt; M I &iacute;&Uuml;&Eacute;&aring; ~ &aring; &lt; &Auml;&aring; K
138. f&Ntilde; M &lt; ~ &lt; &Auml; ~&aring;&Ccedil; &aring; &lt; M I &iacute;&Uuml;&Eacute;&aring; ~ &aring; &gt; &Auml;&aring; K
139. f&Ntilde; M &lt; ~ &lt; &Auml; I &iacute;&Uuml;&Eacute;&aring;
140.
&aring;
~ &lt;&aring; &Auml;K
~+&Auml;
I
O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~ &gt; M I &Auml; &gt; M X ~&aring; &Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute; &aacute;&euml; &icirc;~&auml;&aacute;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde; ~ = &Auml; K
~&Auml; ≤
N
141. ~ + ≥ O I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~ &gt; M X ~&aring; &Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute; &iacute;~&acirc;&Eacute;&euml; &eacute;&auml;~&Aring;&Eacute; &ccedil;&aring;&auml;&oacute; ~&iacute; ~ = N K
~
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CHAPTER 2. ALGEBRA
142.
&aring;
~N~ O K~ &aring; ≤
~N + ~ O + K + ~ &aring;
I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~N I ~ O I KI ~ &aring; &gt; M K
&aring;
&Auml;
143. f&Ntilde; ~&ntilde; + &Auml; &gt; M ~&aring;&Ccedil; ~ &gt; M I &iacute;&Uuml;&Eacute;&aring; &ntilde; &gt; − K
~
&Auml;
144. f&Ntilde; ~&ntilde; + &Auml; &gt; M ~&aring;&Ccedil; ~ &lt; M I &iacute;&Uuml;&Eacute;&aring; &ntilde; &lt; − K
~
145. ~&ntilde; O + &Auml;&ntilde; + &Aring; &gt; M
~&gt;M
~&lt;M
&ntilde; &lt; &ntilde;N I &ntilde; &gt; &ntilde; O
&ntilde;N &lt; &ntilde; &lt; &ntilde; O
&ntilde;N &lt; &ntilde; I &ntilde; &gt; &ntilde;N
&ntilde; ∈∅
−∞&lt; &ntilde; &lt;∞
&ntilde; ∈∅
a&gt;M
a=M
a&lt;M
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CHAPTER 2. ALGEBRA
146.
~+&Auml; ≤ ~ + &Auml;
147. f&Ntilde; &ntilde; &lt; ~ I &iacute;&Uuml;&Eacute;&aring; − ~ &lt; &ntilde; &lt; ~ I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~ &gt; M K
148. f&Ntilde; &ntilde; &gt; ~ I &iacute;&Uuml;&Eacute;&aring; &ntilde; &lt; −~ ~&aring;&Ccedil; &ntilde; &gt; ~ I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~ &gt; M K
149. f&Ntilde; &ntilde; O &lt; ~ I &iacute;&Uuml;&Eacute;&aring; &ntilde; &lt; ~ I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~ &gt; M K
150. f&Ntilde; &ntilde; O &gt; ~ I &iacute;&Uuml;&Eacute;&aring; &ntilde; &gt; ~ I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~ &gt; M K
151. f&Ntilde;
152.
&Ntilde; (&ntilde; )
&gt; M I &iacute;&Uuml;&Eacute;&aring;
&Ouml; (&ntilde; )
&Ntilde; (&ntilde; ) ⋅ &Ouml; (&ntilde; ) &gt; M
K

&Ouml; (&ntilde; ) ≠ M
&Ntilde; (&ntilde; ) ⋅ &Ouml; (&ntilde; ) &lt; M
&Ntilde; (&ntilde; )
K
&lt; M I &iacute;&Uuml;&Eacute;&aring; 
&Ouml; (&ntilde; )
&Ouml; (&ntilde; ) ≠ M
2.8 Compound Interest Formulas
c&igrave;&iacute;&igrave;&ecirc;&Eacute; &icirc;~&auml;&igrave;&Eacute;W ^
f&aring;&aacute;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&eacute;&ccedil;&euml;&aacute;&iacute;W `
^&aring;&aring;&igrave;~&auml; &ecirc;~&iacute;&Eacute; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;W &ecirc;
k&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &oacute;&Eacute;~&ecirc;&euml; &aacute;&aring;&icirc;&Eacute;&euml;&iacute;&Eacute;&Ccedil;W &iacute;
k&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&aacute;&atilde;&Eacute;&euml; &Aring;&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &eacute;&Eacute;&ecirc; &oacute;&Eacute;~&ecirc;W &aring;
153. d&Eacute;&aring;&Eacute;&ecirc;~&auml; `&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil; f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~
&aring;&iacute;
 &ecirc;
^ = ` N + 
 &aring;
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CHAPTER 2. ALGEBRA
154. p&aacute;&atilde;&eacute;&auml;&aacute;&Ntilde;&aacute;&Eacute;&Ccedil; `&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil; f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~
f&Ntilde; &aacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute; &aacute;&euml; &Aring;&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &ccedil;&aring;&Aring;&Eacute; &eacute;&Eacute;&ecirc; &oacute;&Eacute;~&ecirc;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&Eacute;&icirc;&aacute;&ccedil;&igrave;&euml;
&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~ &euml;&aacute;&atilde;&eacute;&auml;&aacute;&Ntilde;&aacute;&Eacute;&euml; &iacute;&ccedil;W
&iacute;
^ = `(N + &ecirc; ) K
155. `&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; `&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil; f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;
f&Ntilde; &aacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute; &aacute;&euml; &Aring;&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;~&auml;&auml;&oacute; E &aring; → ∞ FI &iacute;&Uuml;&Eacute;&aring;
^ = `&Eacute; &ecirc;&iacute; K
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Chapter 3
Geometry
3.1 Right Triangle
i&Eacute;&Ouml;&euml; &ccedil;&Ntilde; ~ &ecirc;&aacute;&Ouml;&Uuml;&iacute; &iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W ~I &Auml;
e&oacute;&eacute;&ccedil;&iacute;&Eacute;&aring;&igrave;&euml;&Eacute;W &Aring;
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W &Uuml;
j&Eacute;&Ccedil;&aacute;~&aring;&euml;W &atilde; ~ I &atilde; &Auml; I &atilde; &Aring;
^&aring;&Ouml;&auml;&Eacute;&euml;W α I β
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
^&ecirc;&Eacute;~W p
Figure 8.
156. α + β = VM&deg;
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CHAPTER 3. GEOMETRY
157. &euml;&aacute;&aring; α =
~
= &Aring;&ccedil;&euml; β
&Aring;
158. &Aring;&ccedil;&euml; α =
&Auml;
= &euml;&aacute;&aring; β
&Aring;
159. &iacute;~&aring; α =
~
= &Aring;&ccedil;&iacute; β
&Auml;
160. &Aring;&ccedil;&iacute; α =
&Auml;
= &iacute;~&aring; β
~
161. &euml;&Eacute;&Aring; α =
&Aring;
= &Aring;&ccedil;&euml; &Eacute;&Aring; β
&Auml;
162. &Aring;&ccedil;&euml; &Eacute;&Aring; α =
&Aring;
= &euml;&Eacute;&Aring; β
~
163. m&oacute;&iacute;&Uuml;~&Ouml;&ccedil;&ecirc;&Eacute;~&aring; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;
~ O + &Auml;O = &Aring; O
164. ~ O = &Ntilde;&Aring; I &Auml;O = &Ouml;&Aring; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Ntilde; ~&aring;&Ccedil; &Aring; ~&ecirc;&Eacute; &eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;&Eacute;&Ouml;&euml; ~ ~&aring;&Ccedil; &Auml;I &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;I &ccedil;&aring;&iacute;&ccedil; &iacute;&Uuml;&Eacute; &Uuml;&oacute;&eacute;&ccedil;&iacute;&Eacute;&aring;&igrave;&euml;&Eacute; &Aring;K
Figure 9.
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CHAPTER 3. GEOMETRY
165. &Uuml; O = &Ntilde;&Ouml; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Uuml; &aacute;&euml; &iacute;&Uuml;&Eacute; ~&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute; &Ntilde;&ecirc;&ccedil;&atilde; &iacute;&Uuml;&Eacute; &ecirc;&aacute;&Ouml;&Uuml;&iacute; ~&aring;&Ouml;&auml;&Eacute;K
~O
&Auml;O
I &atilde; O&Auml; = ~ O − I
Q
Q
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &atilde; ~ ~&aring;&Ccedil; &atilde; &Auml; ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &atilde;&Eacute;&Ccedil;&aacute;~&aring;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &auml;&Eacute;&Ouml;&euml; ~ ~&aring;&Ccedil; &Auml;K
166. &atilde; O~ = &Auml;O −
Figure 10.
&Aring;
167. &atilde; &Aring; = I
O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &atilde; &Aring; &aacute;&euml; &iacute;&Uuml;&Eacute; &atilde;&Eacute;&Ccedil;&aacute;~&aring; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &Uuml;&oacute;&eacute;&ccedil;&iacute;&Eacute;&aring;&igrave;&euml;&Eacute; &Aring;K
168. o =
&Aring;
= &atilde;&Aring;
O
169. &ecirc; =
~ +&Auml;−&Aring;
~&Auml;
=
O
~ + &Auml;+&Aring;
170. ~&Auml; = &Aring;&Uuml;
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CHAPTER 3. GEOMETRY
171. p =
~&Auml; &Aring;&Uuml;
=
O
O
3.2 Isosceles Triangle
_~&euml;&Eacute;W ~
i&Eacute;&Ouml;&euml;W &Auml;
_~&euml;&Eacute; ~&aring;&Ouml;&auml;&Eacute;W β
s&Eacute;&ecirc;&iacute;&Eacute;&ntilde; ~&aring;&Ouml;&auml;&Eacute;W α
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &Auml;~&euml;&Eacute;W &Uuml;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 11.
172. β = VM&deg; −
α
O
~O
173. &Uuml; = &Auml; −
Q
O
O
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CHAPTER 3. GEOMETRY
174. i = ~ + O&Auml;
175. p =
~&Uuml; &Auml;O
= &euml;&aacute;&aring; α
O
O
3.3 Equilateral Triangle
p&aacute;&Ccedil;&Eacute; &ccedil;&Ntilde; ~ &Eacute;&egrave;&igrave;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml; &iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W ~
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W &Uuml;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 12.
176. &Uuml; =
~ P
O
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CHAPTER 3. GEOMETRY
~ P
O
177. o = &Uuml; =
P
P
~ P o
N
178. &ecirc; = &Uuml; =
=
S
O
P
179. i = P~
180. p =
~&Uuml; ~ O P
=
O
Q
3.4 Scalene Triangle
E^ &iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; &iuml;&aacute;&iacute;&Uuml; &aring;&ccedil; &iacute;&iuml;&ccedil; &euml;&aacute;&Ccedil;&Eacute;&euml; &Eacute;&egrave;&igrave;~&auml;F
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; ~ &iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W ~I &Auml;I &Aring;
~ +&Auml;+&Aring;
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &eacute; =
O
^&aring;&Ouml;&auml;&Eacute;&euml; &ccedil;&Ntilde; ~ &iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W αI βI γ
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &euml;&aacute;&Ccedil;&Eacute;&euml; ~I &Auml;I &Aring;W &Uuml; ~ I &Uuml; &Auml; I &Uuml; &Aring;
j&Eacute;&Ccedil;&aacute;~&aring;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &euml;&aacute;&Ccedil;&Eacute;&euml; ~I &Auml;I &Aring;W &atilde; ~ I &atilde; &Auml; I &atilde; &Aring;
_&aacute;&euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; ~&aring;&Ouml;&auml;&Eacute;&euml; αI βI γ W &iacute; ~ I &iacute; &Auml; I &iacute; &Aring;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
^&ecirc;&Eacute;~W p
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CHAPTER 3. GEOMETRY
Figure 13.
181. α + β + γ = NUM&deg;
182. ~ + &Auml; &gt; &Aring; I
&Auml;+&Aring; &gt;~ I
~+&Aring;&gt;&Auml;K
183.
~−&Auml; &lt;&Aring;I
&Auml;−&Aring; &lt;~ I
~−&Aring; &lt;&Auml;K
184. j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;
~
&egrave; = I &egrave; &ouml;&ouml; ~ K
O
Figure 14.
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CHAPTER 3. GEOMETRY
185. i~&iuml; &ccedil;&Ntilde; `&ccedil;&euml;&aacute;&aring;&Eacute;&euml;
~ O = &Auml;O + &Aring; O − O&Auml;&Aring; &Aring;&ccedil;&euml; α I
&Auml;O = ~ O + &Aring; O − O~&Aring; &Aring;&ccedil;&euml; β I
&Aring; O = ~ O + &Auml;O − O~&Auml; &Aring;&ccedil;&euml; γ K
186. i~&iuml; &ccedil;&Ntilde; p&aacute;&aring;&Eacute;&euml;
~
&Auml;
&Aring;
=
=
= Oo I
&euml;&aacute;&aring; α &euml;&aacute;&aring; β &euml;&aacute;&aring; γ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; o &aacute;&euml; &iacute;&Uuml;&Eacute; &ecirc;~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;K
187. o =
~
&Auml;
&Aring;
&Auml;&Aring;
~&Aring;
~&Auml; ~&Auml;&Aring;
=
=
=
=
=
=
O &euml;&aacute;&aring; α O &euml;&aacute;&aring; β O &euml;&aacute;&aring; γ O&Uuml; ~ O&Uuml; &Auml; O&Uuml; &Aring; Qp
188. &ecirc; O =
(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) I
&eacute;
N N
N
N
= +
+ K
&ecirc; &Uuml;~ &Uuml;&Auml; &Uuml;&Aring;
α
=
O
(&eacute; − &Auml;)(&eacute; − &Aring; ) I
&Aring;&ccedil;&euml;
&eacute;(&eacute; − ~ )
α
I
=
O
&Auml;&Aring;
&iacute;~&aring;
α
=
O
(&eacute; − &Auml;)(&eacute; − &Aring; ) K
&eacute;(&eacute; − ~ )
189. &euml;&aacute;&aring;
&Auml;&Aring;
O
&eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) I
~
O
&eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) I
&Uuml;&Auml; =
&Auml;
O
&eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) K
&Uuml;&Aring; =
&Aring;
190. &Uuml; ~ =
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CHAPTER 3. GEOMETRY
191. &Uuml; ~ = &Auml; &euml;&aacute;&aring; γ = &Aring; &euml;&aacute;&aring; β I
&Uuml; &Auml; = ~ &euml;&aacute;&aring; γ = &Aring; &euml;&aacute;&aring; α I
&Uuml; &Aring; = ~ &euml;&aacute;&aring; β = &Auml; &euml;&aacute;&aring; α K
&Auml;O + &Aring; O ~ O
− I
O
Q
O
O
~ + &Aring; &Auml;O
− I
&atilde; O&Auml; =
O
Q
O
O
~ + &Auml; &Aring;O
O
− K
&atilde;&Aring; =
O
Q
192. &atilde; O~ =
Figure 15.
O
O
O
193. ^j = &atilde; ~ I _j = &atilde; &Auml; I `j = &atilde; &Aring; Ec&aacute;&Ouml;KNRFK
P
P
P
Q&Auml;&Aring;&eacute;(&eacute; − ~ )
I
(&Auml; + &Aring; )O
Q~&Aring;&eacute;(&eacute; − &Auml;)
&iacute; O&Auml; =
I
(~ + &Aring; )O
Q~&Auml;&eacute;(&eacute; − &Aring; )
&iacute; O&Aring; =
K
(~ + &Auml;)O
194. &iacute; O~ =
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CHAPTER 3. GEOMETRY
~&Uuml; ~ &Auml;&Uuml; &Auml; &Aring;&Uuml; &Aring;
=
=
I
O
O
O
~&Auml; &euml;&aacute;&aring; γ ~&Aring; &euml;&aacute;&aring; β &Auml;&Aring; &euml;&aacute;&aring; α
I
=
=
p=
O
O
O
p = &eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) Ee&Eacute;&ecirc;&ccedil;&aring;∞&euml; c&ccedil;&ecirc;&atilde;&igrave;&auml;~FI
p = &eacute;&ecirc; I
~&Auml;&Aring;
p=
I
Qo
p = Oo O &euml;&aacute;&aring; α &euml;&aacute;&aring; β &euml;&aacute;&aring; γ I
α
β
γ
p = &eacute;O &iacute;~&aring; &iacute;~&aring; &iacute;~&aring; K
O
O
O
195. p =
3.5 Square
p&aacute;&Ccedil;&Eacute; &ccedil;&Ntilde; ~ &euml;&egrave;&igrave;~&ecirc;&Eacute;W ~
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W &Ccedil;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 16.
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CHAPTER 3. GEOMETRY
196. &Ccedil; = ~ O
197. o =
&Ccedil; ~ O
=
O
O
198. &ecirc; =
~
O
199. i = Q~
200. p = ~ O
3.6 Rectangle
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; ~ &ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;W ~I &Auml;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W &Ccedil;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 17.
201. &Ccedil; = ~ O + &Auml;O
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CHAPTER 3. GEOMETRY
202. o =
&Ccedil;
O
203. i = O(~ + &Auml;)
204. p = ~&Auml;
3.7 Parallelogram
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; ~ &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&ccedil;&Ouml;&ecirc;~&atilde;W ~I &Auml;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W &Ccedil;N I &Ccedil; O
`&ccedil;&aring;&euml;&Eacute;&Aring;&igrave;&iacute;&aacute;&icirc;&Eacute; ~&aring;&Ouml;&auml;&Eacute;&euml;W αI β
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W ϕ
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W &Uuml;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 18.
205. α + β = NUM&deg;
206. &Ccedil;NO + &Ccedil; OO = O(~ O + &Auml;O )
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CHAPTER 3. GEOMETRY
207. &Uuml; = &Auml; &euml;&aacute;&aring; α = &Auml; &euml;&aacute;&aring; β
208. i = O(~ + &Auml;)
209. p = ~&Uuml; = ~&Auml; &euml;&aacute;&aring; α I
N
p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ K
O
3.8 Rhombus
p&aacute;&Ccedil;&Eacute; &ccedil;&Ntilde; ~ &ecirc;&Uuml;&ccedil;&atilde;&Auml;&igrave;&euml;W ~
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W &Ccedil;N I &Ccedil; O
`&ccedil;&aring;&euml;&Eacute;&Aring;&igrave;&iacute;&aacute;&icirc;&Eacute; ~&aring;&Ouml;&auml;&Eacute;&euml;W αI β
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W e
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 19.
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CHAPTER 3. GEOMETRY
210. α + β = NUM&deg;
211. &Ccedil;NO + &Ccedil; OO = Q~ O
212. &Uuml; = ~ &euml;&aacute;&aring; α =
213. &ecirc; =
&Ccedil;N&Ccedil; O
O~
&Uuml; &Ccedil;N&Ccedil; O ~ &euml;&aacute;&aring; α
=
=
O
Q~
O
214. i = Q~
215. p = ~&Uuml; = ~ O &euml;&aacute;&aring; α I
N
p = &Ccedil;N&Ccedil; O K
O
3.9 Trapezoid
_~&euml;&Eacute;&euml; &ccedil;&Ntilde; ~ &iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W ~I &Auml;
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W &egrave;
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W &Uuml;
^&ecirc;&Eacute;~W p
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CHAPTER 3. GEOMETRY
Figure 20.
216. &egrave; =
~+&Auml;
O
217. p =
~+&Auml;
⋅ &Uuml; = &egrave;&Uuml;
O
3.10 Isosceles Trapezoid
_~&euml;&Eacute;&euml; &ccedil;&Ntilde; ~ &iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W ~I &Auml;
i&Eacute;&Ouml;W &Aring;
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W &egrave;
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W &Uuml;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W &Ccedil;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
^&ecirc;&Eacute;~W p
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CHAPTER 3. GEOMETRY
Figure 21.
218. &egrave; =
~+&Auml;
O
219. &Ccedil; = ~&Auml; + &Aring; O
220. &Uuml; = &Aring; O −
221. o =
222. p =
N
(&Auml; − ~ )O
Q
&Aring; ~&Auml; + &Aring; O
(O&Aring; − ~ + &Auml;)(O&Aring; + ~ − &Auml;)
~+&Auml;
⋅ &Uuml; = &egrave;&Uuml;
O
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CHAPTER 3. GEOMETRY
3.11 Isosceles Trapezoid with
Inscribed Circle
_~&euml;&Eacute;&euml; &ccedil;&Ntilde; ~ &iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W ~I &Auml;
i&Eacute;&Ouml;W &Aring;
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W &egrave;
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W &Uuml;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W &Ccedil;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 22.
223. ~ + &Auml; = O&Aring;
224. &egrave; =
~+&Auml;
=&Aring;
O
225. &Ccedil; O = &Uuml; O + &Aring; O
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CHAPTER 3. GEOMETRY
226. &ecirc; =
&Uuml;
~&Auml;
=
O
O
227. o =
&Auml;
~+&Auml; ~
&Aring;O
&Aring;
&Aring;&Ccedil; &Aring;&Ccedil; &Aring;
=
=
=
+S+
&Uuml;O + &Aring; O =
N+
~
U
&Auml;
~&Auml; O&Uuml;
O&Uuml; Q&ecirc; O
228. i = O(~ + &Auml;) = Q&Aring;
229. p =
(~ + &Auml;) ~&Auml; = &egrave;&Uuml; = &Aring;&Uuml; = i&ecirc;
~+&Auml;
⋅&Uuml; =
O
O
O
3.12 Trapezoid with Inscribed Circle
_~&euml;&Eacute;&euml; &ccedil;&Ntilde; ~ &iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W ~I &Auml;
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&aacute;&Ccedil;&Eacute;&euml;W &Aring;I &Ccedil;
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W &egrave;
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W &Uuml;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W &Ccedil;N I &Ccedil; O
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W ϕ
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
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CHAPTER 3. GEOMETRY
Figure 23.
230. ~ + &Auml; = &Aring; + &Ccedil;
231. &egrave; =
~+&Auml; &Aring;+&Ccedil;
=
O
O
232. i = O(~ + &Auml;) = O(&Aring; + &Ccedil; )
&Aring;+&Ccedil;
~+&Auml;
⋅&Uuml; =
⋅ &Uuml; = &egrave;&Uuml; I
O
O
N
p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ K
O
233. p =
3.13 Kite
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; ~ &acirc;&aacute;&iacute;&Eacute;W ~I &Auml;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W &Ccedil;N I &Ccedil; O
^&aring;&Ouml;&auml;&Eacute;&euml;W αI βI γ
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
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CHAPTER 3. GEOMETRY
Figure 24.
234. α + β + Oγ = PSM&deg;
235. i = O(~ + &Auml;)
236. p =
&Ccedil;N&Ccedil; O
O
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; ~ &egrave;&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;W ~I &Auml;I &Aring;I &Ccedil;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W &Ccedil;N I &Ccedil; O
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W ϕ
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml; ~&aring;&Ouml;&auml;&Eacute;&euml;W αI βI γ I δ
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &eacute;
^&ecirc;&Eacute;~W p
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CHAPTER 3. GEOMETRY
Figure 25.
237. α + γ = β + δ = NUM&deg;
238. m&iacute;&ccedil;&auml;&Eacute;&atilde;&oacute;∞&euml; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;
~&Aring; + &Auml;&Ccedil; = &Ccedil;N&Ccedil; O
239. i = ~ + &Auml; + &Aring; + &Ccedil;
240. o =
N
Q
(~&Aring; + &Auml;&Ccedil; )(~&Ccedil; + &Auml;&Aring; )(~&Auml; + &Aring;&Ccedil; ) I
(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; )(&eacute; − &Ccedil; )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &eacute; =
i
K
O
N
241. p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ I
O
p = (&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; )(&eacute; − &Ccedil; ) I
i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &eacute; = K
O
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CHAPTER 3. GEOMETRY
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; ~ &egrave;&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;W ~I &Auml;I &Aring;I &Ccedil;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W &Ccedil;N I &Ccedil; O
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W ϕ
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &eacute;
^&ecirc;&Eacute;~W p
Figure 26.
242. ~ + &Aring; = &Auml; + &Ccedil;
243. i = ~ + &Auml; + &Aring; + &Ccedil; = O(~ + &Aring; ) = O(&Auml; + &Ccedil; )
&Ccedil;NO&Ccedil; OO − (~ − &Auml;) (~ + &Auml; − &eacute; )
I
O&eacute;
i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &eacute; = K
O
O
O
244. &ecirc; =
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CHAPTER 3. GEOMETRY
N
245. p = &eacute;&ecirc; = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ
O
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; ~ &egrave;&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;W ~I &Auml;I &Aring;I &Ccedil;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W &Ccedil;N I &Ccedil; O
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W ϕ
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml; ~&aring;&Ouml;&auml;&Eacute;&euml;W αI βI γ I δ
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 27.
246. α + β + γ + δ = PSM&deg;
247. i = ~ + &Auml; + &Aring; + &Ccedil;
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CHAPTER 3. GEOMETRY
N
248. p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ
O
3.17 Regular Hexagon
p&aacute;&Ccedil;&Eacute;W ~
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml; ~&aring;&Ouml;&auml;&Eacute;W α
p&auml;~&aring;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &atilde;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &eacute;
^&ecirc;&Eacute;~W p
Figure 28.
249. α = NOM&deg;
250. &ecirc; = &atilde; =
~ P
O
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CHAPTER 3. GEOMETRY
251. o = ~
252. i = S~
~OP P
I
O
i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &eacute; = K
O
253. p = &eacute;&ecirc; =
3.18 Regular Polygon
p&aacute;&Ccedil;&Eacute;W ~
k&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &euml;&aacute;&Ccedil;&Eacute;&euml;W &aring;
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml; ~&aring;&Ouml;&auml;&Eacute;W α
p&auml;~&aring;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &atilde;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &eacute;
^&ecirc;&Eacute;~W p
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CHAPTER 3. GEOMETRY
Figure 29.
254. α =
&aring;−O
⋅ NUM&deg;
O
255. α =
&aring;−O
⋅ NUM&deg;
O
~
256. o =
O &euml;&aacute;&aring;
257. &ecirc; = &atilde; =
π
&aring;
~
O &iacute;~&aring;
π
&aring;
= oO −
~O
Q
258. i = &aring;~
259. p =
Oπ
&aring;o O
&euml;&aacute;&aring; I
&aring;
O
p = &eacute;&ecirc; = &eacute; o O −
~O
I
Q
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CHAPTER 3. GEOMETRY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &eacute; =
i
K
O
3.19 Circle
o~&Ccedil;&aacute;&igrave;&euml;W o
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &Ccedil;
`&Uuml;&ccedil;&ecirc;&Ccedil;W ~
p&Eacute;&Aring;~&aring;&iacute; &euml;&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;&euml;W &Eacute;I &Ntilde;
q~&aring;&Ouml;&Eacute;&aring;&iacute; &euml;&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;W &Ouml;
`&Eacute;&aring;&iacute;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute;W α
f&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; ~&aring;&Ouml;&auml;&Eacute;W β
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
260. ~ = Oo &euml;&aacute;&aring;
α
O
Figure 30.
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CHAPTER 3. GEOMETRY
261. ~N~ O = &Auml;N&Auml;O
Figure 31.
262. &Eacute;&Eacute;N = &Ntilde;&Ntilde;N
Figure 32.
263. &Ouml; O = &Ntilde;&Ntilde;N
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CHAPTER 3. GEOMETRY
Figure 33.
264. β =
α
O
Figure 34.
265. i = Oπo = π&Ccedil;
266. p = πo O =
π&Ccedil; O io
=
Q
O
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CHAPTER 3. GEOMETRY
3.20 Sector of a Circle
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; ~ &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
^&ecirc;&Aring; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W &euml;
`&Eacute;&aring;&iacute;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute; E&aacute;&aring; &ecirc;~&Ccedil;&aacute;~&aring;&euml;FW &ntilde;
`&Eacute;&aring;&iacute;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute; E&aacute;&aring; &Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;FW α
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 35.
267. &euml; = o&ntilde;
268. &euml; =
πoα
NUM&deg;
269. i = &euml; + Oo
o&euml; o O &ntilde; πo Oα
270. p =
=
=
O
O
PSM&deg;
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CHAPTER 3. GEOMETRY
3.21 Segment of a Circle
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; ~ &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
^&ecirc;&Aring; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W &euml;
`&Uuml;&ccedil;&ecirc;&Ccedil;W ~
`&Eacute;&aring;&iacute;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute; E&aacute;&aring; &ecirc;~&Ccedil;&aacute;~&aring;&euml;FW &ntilde;
`&Eacute;&aring;&iacute;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute; E&aacute;&aring; &Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;FW α
e&Eacute;&aacute;&Ouml;&Uuml;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;W &Uuml;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
Figure 36.
271. ~ = O O&Uuml;o − &Uuml; O
272. &Uuml; = o −
N
Qo O − ~ O I &Uuml; &lt; o
O
273. i = &euml; + ~
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CHAPTER 3. GEOMETRY
O
O
N
[&euml;o − ~(o − &Uuml; )] = o  απ − &euml;&aacute;&aring; α  = o (&ntilde; − &euml;&aacute;&aring; &ntilde; ) I
O
O  NUM&deg;
 O
O
p ≈ &Uuml;~ K
P
274. p =
3.22 Cube
b&Ccedil;&Ouml;&Eacute;W ~
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W &Ccedil;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W &ecirc;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W &ecirc;
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 37.
275. &Ccedil; = ~ P
276. &ecirc; =
~
O
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CHAPTER 3. GEOMETRY
277. o =
~ P
O
278. p = S~ O
279. s = ~ P
3.23 Rectangular Parallelepiped
b&Ccedil;&Ouml;&Eacute;&euml;W ~I &Auml;I &Aring;
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W &Ccedil;
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 38.
280. &Ccedil; = ~ O + &Auml;O + &Aring; O
281. p = O(~&Auml; + ~&Aring; + &Auml;&Aring; )
282. s = ~&Auml;&Aring;
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CHAPTER 3. GEOMETRY
3.24 Prism
i~&iacute;&Eacute;&ecirc;~&auml; &Eacute;&Ccedil;&Ouml;&Eacute;W &auml;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
i~&iacute;&Eacute;&ecirc;~&auml; ~&ecirc;&Eacute;~W p i
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W p_
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 39.
283. p = p i + Op_ K
284. i~&iacute;&Eacute;&ecirc;~&auml; ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ o&aacute;&Ouml;&Uuml;&iacute; m&ecirc;&aacute;&euml;&atilde;
p i = (~ N + ~ O + ~ P + K + ~ &aring; )&auml;
285. i~&iacute;&Eacute;&ecirc;~&auml; ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~&aring; l&Auml;&auml;&aacute;&egrave;&igrave;&Eacute; m&ecirc;&aacute;&euml;&atilde;
p i = &eacute;&auml; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &eacute; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&ecirc;&ccedil;&euml;&euml; &euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;K
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CHAPTER 3. GEOMETRY
286. s = p_ &Uuml;
287. `~&icirc;~&auml;&aacute;&Eacute;&ecirc;&aacute;D&euml; m&ecirc;&aacute;&aring;&Aring;&aacute;&eacute;&auml;&Eacute;
d&aacute;&icirc;&Eacute;&aring; &iacute;&iuml;&ccedil; &euml;&ccedil;&auml;&aacute;&Ccedil;&euml; &aacute;&aring;&Aring;&auml;&igrave;&Ccedil;&Eacute;&Ccedil; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &eacute;&auml;~&aring;&Eacute;&euml;K f&Ntilde; &Eacute;&icirc;&Eacute;&ecirc;&oacute;
&eacute;&auml;~&aring;&Eacute; &Aring;&ecirc;&ccedil;&euml;&euml; &euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &eacute;&auml;~&aring;&Eacute;&euml; &Uuml;~&euml; &iacute;&Uuml;&Eacute; &euml;~&atilde;&Eacute;
~&ecirc;&Eacute;~ &aacute;&aring; &Auml;&ccedil;&iacute;&Uuml; &euml;&ccedil;&auml;&aacute;&Ccedil;&euml;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &icirc;&ccedil;&auml;&igrave;&atilde;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&ccedil;&auml;&aacute;&Ccedil;&euml; ~&ecirc;&Eacute; &Eacute;&egrave;&igrave;~&auml;K
3.25 Regular Tetrahedron
q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; &euml;&aacute;&Ccedil;&Eacute; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W ~
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W p_
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 40.
288. &Uuml; =
O
~
P
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CHAPTER 3. GEOMETRY
289. p_ =
P~ O
Q
290. p = P~ O
~P
N
291. s = p_ &Uuml; =
K
P
S O
3.26 Regular Pyramid
p&aacute;&Ccedil;&Eacute; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W ~
i~&iacute;&Eacute;&ecirc;~&auml; &Eacute;&Ccedil;&Ouml;&Eacute;W &Auml;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
p&auml;~&aring;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &atilde;
k&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &euml;&aacute;&Ccedil;&Eacute;&euml;W &aring;
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W &eacute;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W &ecirc;
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W p_
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W pi
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 41.
292. &atilde; = &Auml;O −
293. &Uuml; =
~O
Q
π O
−~
&aring;
π
O &euml;&aacute;&aring;
&aring;
Q&Auml;O &euml;&aacute;&aring; O
N
N
294. p i = &aring;~&atilde; = &aring;~ Q&Auml;O − ~ O = &eacute;&atilde;
Q
O
295. p_ = &eacute;&ecirc;
296. p = p_ + p i
N
N
297. s = p_ &Uuml; = &eacute;&ecirc;&Uuml;
P
P
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CHAPTER 3. GEOMETRY
3.27 Frustum of a Regular Pyramid
~N I ~ O I ~ P IKI ~ &aring;
_~&euml;&Eacute; ~&aring;&Ccedil; &iacute;&ccedil;&eacute; &euml;&aacute;&Ccedil;&Eacute; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;&euml;W 
&Auml;N I &Auml;O I &Auml;P IKI &Auml;&aring;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
p&auml;~&aring;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &atilde;
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;&euml;W pN I pO
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p i
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;&euml;W mN I mO
p&Aring;~&auml;&Eacute; &Ntilde;~&Aring;&iacute;&ccedil;&ecirc;W &acirc;
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 42.
298.
&Auml;N &Auml;O &Auml;P
&Auml;
&Auml;
= = =K= &aring; = = &acirc;
~N ~ O ~ P
~&aring; ~
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CHAPTER 3. GEOMETRY
299.
pO
= &acirc;O
pN
300. p i =
&atilde;(mN + mO )
O
301. p = p i + pN + pO
(
)
302. s =
&Uuml;
pN + pNpO + pO
P
303. s =
O
&Uuml;pN  &Auml;  &Auml;   &Uuml;pN
N+ &acirc; + &acirc;O
N
+
+
  =

P  ~ ~  P
[
]
3.28 Rectangular Right Wedge
p&aacute;&Ccedil;&Eacute;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W ~I &Auml;
q&ccedil;&eacute; &Eacute;&Ccedil;&Ouml;&Eacute;W &Aring;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p i
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W p_
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 43.
304. p i =
N
(~ + &Aring; ) Q&Uuml;O + &Auml;O + &Auml; &Uuml;O + (~ − &Aring; )O
O
305. p_ = ~&Auml;
306. p = p_ + p i
307. s =
&Auml;&Uuml;
(O~ + &Aring; )
S
3.29 Platonic Solids
b&Ccedil;&Ouml;&Eacute;W ~
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W &ecirc;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W o
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
308. c&aacute;&icirc;&Eacute; m&auml;~&iacute;&ccedil;&aring;&aacute;&Aring; p&ccedil;&auml;&aacute;&Ccedil;&euml;
q&Uuml;&Eacute; &eacute;&auml;~&iacute;&ccedil;&aring;&aacute;&Aring; &euml;&ccedil;&auml;&aacute;&Ccedil;&euml; ~&ecirc;&Eacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ntilde; &eacute;&ccedil;&auml;&oacute;&Uuml;&Eacute;&Ccedil;&ecirc;~ &iuml;&aacute;&iacute;&Uuml; &Eacute;&egrave;&igrave;&aacute;&icirc;~&auml;&Eacute;&aring;&iacute;
&Ntilde;~&Aring;&Eacute;&euml; &Aring;&ccedil;&atilde;&eacute;&ccedil;&euml;&Eacute;&Ccedil; &ccedil;&Ntilde; &Aring;&ccedil;&aring;&Ouml;&ecirc;&igrave;&Eacute;&aring;&iacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ntilde; &ecirc;&Eacute;&Ouml;&igrave;&auml;~&ecirc; &eacute;&ccedil;&auml;&oacute;&Ouml;&ccedil;&aring;&euml;K
p&ccedil;&auml;&aacute;&Ccedil;
k&igrave;&atilde;&Auml;&Eacute;&ecirc;
&ccedil;&Ntilde; s&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml;
q&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;
Q
`&igrave;&Auml;&Eacute;
U
l&Aring;&iacute;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;
S
f&Aring;&ccedil;&euml;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;
NO
a&ccedil;&Ccedil;&Eacute;&Aring;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;
OM
k&igrave;&atilde;&Auml;&Eacute;&ecirc;
&ccedil;&Ntilde; b&Ccedil;&Ouml;&Eacute;&euml;
S
NO
NO
PM
PM
k&igrave;&atilde;&Auml;&Eacute;&ecirc;
&ccedil;&Ntilde; c~&Aring;&Eacute;&euml;
Q
S
U
OM
NO
Octahedron
Figure 44.
309. &ecirc; =
~ S
S
310. o =
~ O
O
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p&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;
PKOR
PKOO
PKOT
PKOT
PKOT
CHAPTER 3. GEOMETRY
311. p = O~ O P
312. s =
~P O
P
Icosahedron
Figure 45.
(
)
313. &ecirc; =
~ P P+ R
NO
314. o =
~
O R+ R
Q
(
)
315. p = R~ O P
(
R~ P P + R
316. s =
NO
)
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CHAPTER 3. GEOMETRY
Dodecahedron
Figure 46.
(
317. &ecirc; =
~ NM OR + NN R
O
318. o =
~ P N+ R
Q
(
)
(
319. p = P~ O R R + O R
320. s =
(
)
~ P NR + T R
Q
)
)
3.30 Right Circular Cylinder
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W o
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W &Ccedil;
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CHAPTER 3. GEOMETRY
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W e
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p i
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W p_
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 47.
321. p i = Oπoe
&Ccedil;

322. p = p i + Op_ = Oπo(e + o ) = π&Ccedil; e + 
O

323. s = p_ e = πo O e
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CHAPTER 3. GEOMETRY
3.31 Right Circular Cylinder with
an Oblique Plane Face
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W o
q&Uuml;&Eacute; &Ouml;&ecirc;&Eacute;~&iacute;&Eacute;&euml;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute; &ccedil;&Ntilde; ~ &euml;&aacute;&Ccedil;&Eacute;W &Uuml;N
q&Uuml;&Eacute; &euml;&Uuml;&ccedil;&ecirc;&iacute;&Eacute;&euml;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute; &ccedil;&Ntilde; ~ &euml;&aacute;&Ccedil;&Eacute;W &Uuml; O
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p i
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &eacute;&auml;~&aring;&Eacute; &Eacute;&aring;&Ccedil; &Ntilde;~&Aring;&Eacute;&euml;W p_
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 48.
324. p i = πo(&Uuml;N + &Uuml; O )
 &Uuml; − &Uuml;O 
325. p_ = πo + πo o +  N

 O 
O
O
O
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CHAPTER 3. GEOMETRY
O

&Uuml;N − &Uuml; O  

O
326. p = p i + p_ = πo &Uuml;N + &Uuml; O + o + o + 
 
 O  

327. s =
πo O
(&Uuml;N + &Uuml;O )
O
3.32 Right Circular Cone
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W o
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W &Ccedil;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W e
p&auml;~&aring;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &atilde;
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W pi
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W p_
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 49.
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CHAPTER 3. GEOMETRY
328. e = &atilde; O − o O
329. p i = πo&atilde; =
π&atilde;&Ccedil;
O
330. p_ = πo O
&Ccedil;
N 
331. p = p i + p_ = πo (&atilde; + o ) = π&Ccedil; &atilde; + 
O
O 
N
N
332. s = p_ e = πo O e
P
P
3.33 Frustum of a Right Circular Cone
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;&euml;W oI &ecirc;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W e
p&auml;~&aring;&iacute; &Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &atilde;
p&Aring;~&auml;&Eacute; &Ntilde;~&Aring;&iacute;&ccedil;&ecirc;W &acirc;
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &Auml;~&euml;&Eacute;&euml;W pN I pO
i~&iacute;&Eacute;&ecirc;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p i
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 50.
333. e = &atilde; O − (o − &ecirc; )
O
334.
o
=&acirc;
&ecirc;
335.
pO o O
= O = &acirc;O
pN &ecirc;
336. p i = π&atilde;(o + &ecirc; )
[
]
337. p = pN + pO + p i = π o O + &ecirc; O + &atilde;(o + &ecirc; )
338. s =
(
&Uuml;
pN + pNpO + pO
P
)
O
&Uuml;pN  o  o   &Uuml;pN
339. s =
N+ &acirc; + &acirc;O
N + +    =
P  &ecirc; &ecirc;  P
[
]
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CHAPTER 3. GEOMETRY
3.34 Sphere
o~&Ccedil;&aacute;&igrave;&euml;W o
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &Ccedil;
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
Figure 51.
340. p = Qπo O
Q
N
N
341. s = πo P e = π&Ccedil; P = po
P
S
P
3.35 Spherical Cap
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;W &ecirc;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &eacute;&auml;~&aring;&Eacute; &Ntilde;~&Aring;&Eacute;W p_
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; &Aring;~&eacute;W p`
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 52.
&ecirc; O + &Uuml;O
342. o =
O&Uuml;
343. p_ = π&ecirc; O
344. p` = π(&Uuml; O + &ecirc; O )
345. p = p_ + p` = π(&Uuml; O + O&ecirc; O ) = π(Oo&Uuml; + &ecirc; O )
346. s =
π O
π
&Uuml; (Po − &Uuml; ) = &Uuml;(P&ecirc; O + &Uuml; O )
S
S
3.36 Spherical Sector
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute; &ccedil;&Ntilde; &euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; &Aring;~&eacute;W &ecirc;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 53.
347. p = πo(O&Uuml; + &ecirc; )
O
348. s = πo O &Uuml;
P
k&ccedil;&iacute;&Eacute;W q&Uuml;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; ~&ecirc;&Eacute; &Aring;&ccedil;&ecirc;&ecirc;&Eacute;&Aring;&iacute; &Auml;&ccedil;&iacute;&Uuml; &Ntilde;&ccedil;&ecirc; &plusmn;&ccedil;&eacute;&Eacute;&aring;≤ ~&aring;&Ccedil;
&plusmn;&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;≤ &euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; &euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;K
3.37 Spherical Segment
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W o
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Auml;~&euml;&Eacute;&euml;W &ecirc;N I &ecirc;O
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W &Uuml;
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W pp
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &eacute;&auml;~&aring;&Eacute; &Eacute;&aring;&Ccedil; &Ntilde;~&Aring;&Eacute;&euml;W pN I pO
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 54.
349. pp = Oπo&Uuml;
350. p = pp + pN + pO = π(Oo&Uuml; + &ecirc;NO + &ecirc;OO )
N
351. s = π&Uuml;(P&ecirc;NO + P&ecirc;OO + &Uuml; O )
S
3.38 Spherical Wedge
o~&Ccedil;&aacute;&igrave;&euml;W o
a&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute; &aacute;&aring; &Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;W &ntilde;
a&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute; &aacute;&aring; &ecirc;~&Ccedil;&aacute;~&aring;&euml;W α
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; &auml;&igrave;&aring;&Eacute;W p i
q&ccedil;&iacute;~&auml; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 55.
352. p i =
πo O
α = Oo O &ntilde;
VM
353. p = πo O +
354. s =
πo O
α = πo O + Oo O &ntilde;
VM
πoP
O
α = oP &ntilde;
OTM
P
3.39 Ellipsoid
p&Eacute;&atilde;&aacute;-~&ntilde;&Eacute;&euml;W ~I &Auml;I &Aring;
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Figure 56.
Q
355. s = π~&Auml;&Aring;
P
Prolate Spheroid
p&Eacute;&atilde;&aacute;-~&ntilde;&Eacute;&euml;W ~I &Auml;I &Auml; E ~ &gt; &Auml; F
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
~ ~&ecirc;&Aring;&euml;&aacute;&aring; &Eacute; 

356. p = Oπ&Auml; &Auml; +
I
&Eacute;


&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Eacute; =
~ O − &Auml;O
K
~
Q
357. s = π&Auml;O~
P
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CHAPTER 3. GEOMETRY
Oblate Spheroid
p&Eacute;&atilde;&aacute;-~&ntilde;&Eacute;&euml;W ~I &Auml;I &Auml; E ~ &lt; &Auml; F
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s

 &Auml;&Eacute;  
~ ~&ecirc;&Aring;&euml;&aacute;&aring;&Uuml;   

 ~ I
358. p = Oπ&Auml; &Auml; +


&Auml;&Eacute; L ~




&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Eacute; =
&Auml;O − ~ O
K
&Auml;
Q
359. s = π&Auml;O~
P
3.40 Circular Torus
j~&agrave;&ccedil;&ecirc; &ecirc;~&Ccedil;&aacute;&igrave;&euml;W o
j&aacute;&aring;&ccedil;&ecirc; &ecirc;~&Ccedil;&aacute;&igrave;&euml;W &ecirc;
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 3. GEOMETRY
Picture 57.
360. p = QπOo&ecirc;
361. s = OπOo&ecirc; O
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Chapter 4
Trigonometry
^&aring;&Ouml;&auml;&Eacute;&euml;W α I β
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; E&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; ~ &eacute;&ccedil;&aacute;&aring;&iacute;FW &ntilde;I &oacute;
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &acirc;
4.1 Radian and Degree Measures of Angles
362. N &ecirc;~&Ccedil; =
NUM&deg;
≈ RT&deg;NT DQR?
π
363. N&deg; =
π
&ecirc;~&Ccedil; ≈ MKMNTQRP &ecirc;~&Ccedil;
NUM
364. N D =
π
&ecirc;~&Ccedil; ≈ MKMMMOVN &ecirc;~&Ccedil;
NUM ⋅ SM
365. N ? =
π
&ecirc;~&Ccedil; ≈ MKMMMMMR &ecirc;~&Ccedil;
NUM ⋅ PSMM
366.
^&aring;&Ouml;&auml;&Eacute;
E&Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;F
^&aring;&Ouml;&auml;&Eacute;
E&ecirc;~&Ccedil;&aacute;~&aring;&euml;F
M
PM
QR
SM
VM
NUM
OTM
PSM
M
π
S
π
Q
π
P
π
O
π
Pπ
O
Oπ
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CHAPTER 4. TRIGONOMETRY
4.2 Definitions and Graphs of Trigonometric
Functions
Figure 58.
367. &euml;&aacute;&aring; α =
&oacute;
&ecirc;
368. &Aring;&ccedil;&euml; α =
&ntilde;
&ecirc;
369. &iacute;~&aring; α =
&oacute;
&ntilde;
370. &Aring;&ccedil;&iacute; α =
&ntilde;
&oacute;
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CHAPTER 4. TRIGONOMETRY
371. &euml;&Eacute;&Aring; α =
&ecirc;
&ntilde;
372. &Aring;&ccedil;&euml;&Eacute;&Aring; α =
&ecirc;
&oacute;
373. p&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &euml;&aacute;&aring; &ntilde; I − N ≤ &euml;&aacute;&aring; &ntilde; ≤ N K
Figure 59.
374. `&ccedil;&euml;&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &Aring;&ccedil;&euml; &ntilde; I − N ≤ &Aring;&ccedil;&euml; &ntilde; ≤ N K
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CHAPTER 4. TRIGONOMETRY
Figure 60.
375. q~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
π
&oacute; = &iacute;~&aring; &ntilde; I &ntilde; ≠ (O&acirc; + N) I − ∞ ≤ &iacute;~&aring; &ntilde; ≤ ∞K
O
Figure 61.
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CHAPTER 4. TRIGONOMETRY
376. `&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &Aring;&ccedil;&iacute; &ntilde; I &ntilde; ≠ &acirc;π I − ∞ ≤ &Aring;&ccedil;&iacute; &ntilde; ≤ ∞ K
Figure 62.
377. p&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
π
&oacute; = &euml;&Eacute;&Aring; &ntilde; I &ntilde; ≠ (O&acirc; + N) K
O
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CHAPTER 4. TRIGONOMETRY
Figure 63.
378. `&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &Aring;&ccedil;&euml; &Eacute;&Aring; &ntilde; I &ntilde; ≠ &acirc;π K
Figure 64.
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CHAPTER 4. TRIGONOMETRY
4.3. Signs of Trigonometric Functions
379.
n&igrave;~&Ccedil;&ecirc;~&aring;&iacute;
f
ff
fff
fs
p&aacute;&aring;
α
H
H
`&ccedil;&euml;
α
H
q~&aring;
α
H
`&ccedil;&iacute;
α
H
H
H
H
380.
Figure 65.
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p&Eacute;&Aring;
α
H
H
`&ccedil;&euml;&Eacute;&Aring;
α
H
H
CHAPTER 4. TRIGONOMETRY
4.4 Trigonometric Functions of Common
Angles
381.
α&deg; α &ecirc;~&Ccedil;
M
M
π
PM
S
π
QR
Q
π
SM
P
π
VM
O
Oπ
NOM
P
NUM
π
Pπ
OTM
O
PSM
Oπ
&iacute;~&aring; α
M
N
P
&Aring;&ccedil;&iacute; α
O
O
P
O
&Aring;&ccedil;&euml; α
N
P
O
O
O
N
O
N
N
P
N
P
O
O
P
N
M
∞
M
∞
N
&euml;&aacute;&aring; α
M
N
O
P
O
M
N
O
−N
−
∞
P
&euml;&Eacute;&Aring; α
N
O
P
O
&Aring;&ccedil;&euml;&Eacute;&Aring; α
∞
O
O
M
N
P
∞
−N
O
P
∞
− P
−
−O
−N
M
∞
M
∞
−N
M
N
M
∞
N
∞
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CHAPTER 4. TRIGONOMETRY
382.
α &ecirc;~&Ccedil;
α&deg;
π
NR
NO
&euml;&aacute;&aring; α
&Aring;&ccedil;&euml; α
&iacute;~&aring; α
&Aring;&ccedil;&iacute; α
S− O
Q
S+ O
Q
O− P
O+ P
R+O R
NU
π
NM
R −N
Q
NM + O R
Q
R−O R
R
PS
π
R
NM − O R
Q
R +N
Q
NM − O R
R +N
NM − O R
RQ
Pπ
NM
R +N
Q
NM − O R
Q
R +N
NM − O R
R +N
TO
Oπ
R
NM + O R
Q
R −N
Q
R+O R
R−O R
R
TR
Rπ
NO
S+ O
Q
S− O
Q
O+ P
O− P
NM − O R
4.5 Most Important Formulas
383. &euml;&aacute;&aring; O α + &Aring;&ccedil;&euml; O α = N
384. &euml;&Eacute;&Aring; O α − &iacute;~&aring; O α = N
385. &Aring;&euml;&Aring; O α − &Aring;&ccedil;&iacute; O α = N
386. &iacute;~&aring; α =
&euml;&aacute;&aring; α
&Aring;&ccedil;&euml; α
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R +N
CHAPTER 4. TRIGONOMETRY
387. &Aring;&ccedil;&iacute; α =
&Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
388. &iacute;~&aring; α ⋅ &Aring;&ccedil;&iacute; α = N
389. &euml;&Eacute;&Aring; α =
N
&Aring;&ccedil;&euml; α
390. &Aring;&ccedil;&euml;&Eacute;&Aring; α =
N
&euml;&aacute;&aring; α
4.6 Reduction Formulas
391.
β
−α
VM&deg; − α
VM&deg; + α
NUM&deg; − α
NUM&deg; + α
OTM&deg; − α
OTM&deg; + α
PSM&deg; − α
PSM&deg; + α
&euml;&aacute;&aring; β
− &euml;&aacute;&aring; α
+ &Aring;&ccedil;&euml; α
+ &Aring;&ccedil;&euml; α
+ &euml;&aacute;&aring; α
− &euml;&aacute;&aring; α
− &Aring;&ccedil;&euml; α
− &Aring;&ccedil;&euml; α
− &euml;&aacute;&aring; α
+ &euml;&aacute;&aring; α
&Aring;&ccedil;&euml; β
+ &Aring;&ccedil;&euml; α
+ &euml;&aacute;&aring; α
− &euml;&aacute;&aring; α
− &Aring;&ccedil;&euml; α
− &Aring;&ccedil;&euml; α
− &euml;&aacute;&aring; α
+ &euml;&aacute;&aring; α
+ &Aring;&ccedil;&euml; α
+ &Aring;&ccedil;&euml; α
&iacute;~&aring; β
− &iacute;~&aring; α
+ &Aring;&ccedil;&iacute; α
− &Aring;&ccedil;&iacute; α
− &iacute;~&aring; α
+ &iacute;~&aring; α
+ &Aring;&ccedil;&iacute; α
− &Aring;&ccedil;&iacute; α
− &iacute;~&aring; α
+ &iacute;~&aring; α
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&Aring;&ccedil;&iacute; β
− &Aring;&ccedil;&iacute; α
+ &iacute;~&aring; α
− &iacute;~&aring; α
− &Aring;&ccedil;&iacute; α
+ &Aring;&ccedil;&iacute; α
+ &iacute;~&aring; α
− &iacute;~&aring; α
− &Aring;&ccedil;&iacute; α
+ &Aring;&ccedil;&iacute; α
CHAPTER 4. TRIGONOMETRY
4.7 Periodicity of Trigonometric Functions
392. &euml;&aacute;&aring;(α &plusmn; Oπ&aring; ) = &euml;&aacute;&aring; α I &eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil; Oπ &ccedil;&ecirc; PSM&deg; K
393. &Aring;&ccedil;&euml;(α &plusmn; Oπ&aring; ) = &Aring;&ccedil;&euml; α I &eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil; Oπ &ccedil;&ecirc; PSM&deg; K
394. &iacute;~&aring;(α &plusmn; π&aring; ) = &iacute;~&aring; α I &eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil; π &ccedil;&ecirc; NUM&deg; K
395. &Aring;&ccedil;&iacute;(α &plusmn; π&aring; ) = &Aring;&ccedil;&iacute; α I &eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil; π &ccedil;&ecirc; NUM&deg; K
4.8 Relations between Trigonometric
Functions
396. &euml;&aacute;&aring; α = &plusmn; N − &Aring;&ccedil;&euml; O α = &plusmn;
α
O
=
α
N + &iacute;~&aring; O
O
N
(N − &Aring;&ccedil;&euml; Oα ) = O &Aring;&ccedil;&euml; O  α − π  − N
O
 O Q
O &iacute;~&aring;
397. &Aring;&ccedil;&euml; α = &plusmn; N − &euml;&aacute;&aring; O α = &plusmn;
α
O
=
α
N + &iacute;~&aring; O
O
N
(N + &Aring;&ccedil;&euml; Oα ) = O &Aring;&ccedil;&euml; O α − N
O
O
N − &iacute;~&aring; O
398. &iacute;~&aring; α =
&euml;&aacute;&aring; α
&euml;&aacute;&aring; Oα
N − &Aring;&ccedil;&euml; Oα
= &plusmn; &euml;&Eacute;&Aring; O α − N =
=
&Aring;&ccedil;&euml; α
N + &Aring;&ccedil;&euml; Oα
&euml;&aacute;&aring; Oα
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α
N − &Aring;&ccedil;&euml; Oα
O
=&plusmn;
=
N + &Aring;&ccedil;&euml; Oα
O α
N + &iacute;~&aring;
O
O &iacute;~&aring;
399. &Aring;&ccedil;&iacute; α =
&Aring;&ccedil;&euml; α
N + &Aring;&ccedil;&euml; Oα
&euml;&aacute;&aring; Oα
= &plusmn; &Aring;&euml;&Aring; O α − N =
=
&euml;&aacute;&aring; α
&euml;&aacute;&aring; Oα
N − &Aring;&ccedil;&euml; Oα
α
N − &iacute;~&aring; O
N + &Aring;&ccedil;&euml; Oα
O
=&plusmn;
=
α
N − &Aring;&ccedil;&euml; Oα
O &iacute;~&aring;
O
α
N
O
400. &euml;&Eacute;&Aring; α =
= &plusmn; N + &iacute;~&aring; O α =
α
&Aring;&ccedil;&euml; α
N − &iacute;~&aring; O
O
N + &iacute;~&aring; O
N
401. &Aring;&euml;&Aring; α =
= &plusmn; N + &Aring;&ccedil;&iacute; O α =
&euml;&aacute;&aring; α
α
O
α
O &iacute;~&aring;
O
N + &iacute;~&aring; O
4.9 Addition and Subtraction Formulas
402. &euml;&aacute;&aring;(α + β) = &euml;&aacute;&aring; α &Aring;&ccedil;&euml; β + &euml;&aacute;&aring; β &Aring;&ccedil;&euml; α
403. &euml;&aacute;&aring;(α − &oacute; ) = &euml;&aacute;&aring; α &Aring;&ccedil;&euml; β − &euml;&aacute;&aring; β &Aring;&ccedil;&euml; α
404. &Aring;&ccedil;&euml;(α + β ) = &Aring;&ccedil;&euml; α &Aring;&ccedil;&euml; β − &euml;&aacute;&aring; α &euml;&aacute;&aring; β
405. &Aring;&ccedil;&euml;(α − β ) = &Aring;&ccedil;&euml; α &Aring;&ccedil;&euml; β + &euml;&aacute;&aring; α &euml;&aacute;&aring; β
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CHAPTER 4. TRIGONOMETRY
406. &iacute;~&aring;(α + β ) =
&iacute;~&aring; α + &iacute;~&aring; β
N − &iacute;~&aring; α &iacute;~&aring; β
407. &iacute;~&aring;(α − β ) =
&iacute;~&aring; α − &iacute;~&aring; β
N + &iacute;~&aring; α &iacute;~&aring; β
408. &Aring;&ccedil;&iacute;(α + β) =
N − &iacute;~&aring; α &iacute;~&aring; β
&iacute;~&aring; α + &iacute;~&aring; β
409. &Aring;&ccedil;&iacute;(α − β) =
N + &iacute;~&aring; α &iacute;~&aring; β
&iacute;~&aring; α − &iacute;~&aring; β
4.10 Double Angle Formulas
410. &euml;&aacute;&aring; Oα = O &euml;&aacute;&aring; α ⋅ &Aring;&ccedil;&euml; α
411. &Aring;&ccedil;&euml; Oα = &Aring;&ccedil;&euml; O α − &euml;&aacute;&aring; O α = N − O &euml;&aacute;&aring; O α = O &Aring;&ccedil;&euml; O α − N
412. &iacute;~&aring; Oα =
O &iacute;~&aring; α
O
=
O
N − &iacute;~&aring; α &Aring;&ccedil;&iacute; α − &iacute;~&aring; α
&Aring;&ccedil;&iacute; O α − N &Aring;&ccedil;&iacute; α − &iacute;~&aring; α
413. &Aring;&ccedil;&iacute; Oα =
=
O &Aring;&ccedil;&iacute; α
O
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CHAPTER 4. TRIGONOMETRY
4.11 Multiple Angle Formulas
414. &euml;&aacute;&aring; Pα = P &euml;&aacute;&aring; α − Q &euml;&aacute;&aring; P α = P &Aring;&ccedil;&euml; O α ⋅ &euml;&aacute;&aring; α − &euml;&aacute;&aring;P α
415. &euml;&aacute;&aring; Qα = Q &euml;&aacute;&aring; α ⋅ &Aring;&ccedil;&euml; α − U &euml;&aacute;&aring; P α ⋅ &Aring;&ccedil;&euml; α
416. &euml;&aacute;&aring; Rα = R &euml;&aacute;&aring; α − OM &euml;&aacute;&aring; P α + NS &euml;&aacute;&aring; R α
417. &Aring;&ccedil;&euml; Pα = Q &Aring;&ccedil;&euml;P α − P &Aring;&ccedil;&euml; α = &Aring;&ccedil;&euml; P α − P &Aring;&ccedil;&euml; α ⋅ &euml;&aacute;&aring; O α
418. &Aring;&ccedil;&euml; Qα = U &Aring;&ccedil;&euml; Q α − U &Aring;&ccedil;&euml; O α + N
419. &Aring;&ccedil;&euml; Rα = NS &Aring;&ccedil;&euml; R α − OM &Aring;&ccedil;&euml; P α + R &Aring;&ccedil;&euml; α
420. &iacute;~&aring; Pα =
P &iacute;~&aring; α − &iacute;~&aring; P α
N − P &iacute;~&aring; O α
Q &iacute;~&aring; α − Q &iacute;~&aring; P α
421. &iacute;~&aring; Qα =
N − S &iacute;~&aring; O α + &iacute;~&aring; Q α
422. &iacute;~&aring; Rα =
&iacute;~&aring; R α − NM &iacute;~&aring; P α + R &iacute;~&aring; α
N − NM &iacute;~&aring; O α + R &iacute;~&aring; Q α
423. &Aring;&ccedil;&iacute; Pα =
&Aring;&ccedil;&iacute; P α − P &Aring;&ccedil;&iacute; α
P &Aring;&ccedil;&iacute; O α − N
424. &Aring;&ccedil;&iacute; Qα =
N − S &iacute;~&aring; O α + &iacute;~&aring; Q α
Q &iacute;~&aring; α − Q &iacute;~&aring; P α
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CHAPTER 4. TRIGONOMETRY
425. &Aring;&ccedil;&iacute; Rα =
N − NM &iacute;~&aring; O α + R &iacute;~&aring; Q α
&iacute;~&aring; R α − NM &iacute;~&aring; P α + R &iacute;~&aring; α
4.12 Half Angle Formulas
426. &euml;&aacute;&aring;
α
N − &Aring;&ccedil;&euml; α
=&plusmn;
O
O
427. &Aring;&ccedil;&euml;
α
N + &Aring;&ccedil;&euml; α
=&plusmn;
O
O
428. &iacute;~&aring;
α
N − &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
N − &Aring;&ccedil;&euml; α
=&plusmn;
=
=
= &Aring;&euml;&Aring; α − &Aring;&ccedil;&iacute; α
O
N + &Aring;&ccedil;&euml; α N + &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
429. &Aring;&ccedil;&iacute;
α
N + &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
N + &Aring;&ccedil;&euml; α
=&plusmn;
=
=
= &Aring;&euml;&Aring; α + &Aring;&ccedil;&iacute; α
O
N − &Aring;&ccedil;&euml; α N − &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
4.13 Half Angle Tangent Identities
α
O
430. &euml;&aacute;&aring; α =
α
N + &iacute;~&aring; O
O
O &iacute;~&aring;
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CHAPTER 4. TRIGONOMETRY
α
O
431. &Aring;&ccedil;&euml; α =
O α
N + &iacute;~&aring;
O
N − &iacute;~&aring; O
α
O
432. &iacute;~&aring; α =
α
N − &iacute;~&aring; O
O
O &iacute;~&aring;
α
O
433. &Aring;&ccedil;&iacute; α =
α
O &iacute;~&aring;
O
N − &iacute;~&aring; O
4.14 Transforming of Trigonometric
Expressions to Product
434. &euml;&aacute;&aring; α + &euml;&aacute;&aring; β = O &euml;&aacute;&aring;
α+β
α −β
&Aring;&ccedil;&euml;
O
O
435. &euml;&aacute;&aring; α − &euml;&aacute;&aring; β = O &Aring;&ccedil;&euml;
α +β
α −β
&euml;&aacute;&aring;
O
O
436. &Aring;&ccedil;&euml; α + &Aring;&ccedil;&euml; β = O &Aring;&ccedil;&euml;
α+β
α −β
&Aring;&ccedil;&euml;
O
O
437. &Aring;&ccedil;&euml; α − &Aring;&ccedil;&euml; β = −O &euml;&aacute;&aring;
α +β
α −β
&euml;&aacute;&aring;
O
O
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CHAPTER 4. TRIGONOMETRY
438. &iacute;~&aring; α + &iacute;~&aring; β =
&euml;&aacute;&aring;(α + β )
&Aring;&ccedil;&euml; α ⋅ &Aring;&ccedil;&euml; β
439. &iacute;~&aring; α − &iacute;~&aring; β =
&euml;&aacute;&aring;(α − β )
&Aring;&ccedil;&euml; α ⋅ &Aring;&ccedil;&euml; β
440. &Aring;&ccedil;&iacute; α + &Aring;&ccedil;&iacute; β =
&euml;&aacute;&aring;(β + α )
&euml;&aacute;&aring; α ⋅ &euml;&aacute;&aring; β
441. &Aring;&ccedil;&iacute; α − &Aring;&ccedil;&iacute; β =
&euml;&aacute;&aring;(β − α )
&euml;&aacute;&aring; α ⋅ &euml;&aacute;&aring; β
π

π

442. &Aring;&ccedil;&euml; α + &euml;&aacute;&aring; α = O &Aring;&ccedil;&euml; − α  = O &euml;&aacute;&aring; + α 
Q

Q

π

π

443. &Aring;&ccedil;&euml; α − &euml;&aacute;&aring; α = O &euml;&aacute;&aring; − α  = O &Aring;&ccedil;&euml; + α 
Q

Q

444. &iacute;~&aring; α + &Aring;&ccedil;&iacute; β =
&Aring;&ccedil;&euml;(α − β)
&Aring;&ccedil;&euml; α ⋅ &euml;&aacute;&aring; β
445. &iacute;~&aring; α − &Aring;&ccedil;&iacute; β = −
&Aring;&ccedil;&euml;(α + β )
&Aring;&ccedil;&euml; α ⋅ &euml;&aacute;&aring; β
446. N + &Aring;&ccedil;&euml; α = O &Aring;&ccedil;&euml; O
α
O
447. N − &Aring;&ccedil;&euml; α = O &euml;&aacute;&aring; O
α
O
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CHAPTER 4. TRIGONOMETRY
π α
448. N + &euml;&aacute;&aring; α = O &Aring;&ccedil;&euml; O  − 
Q O
π α
449. N − &euml;&aacute;&aring; α = O &euml;&aacute;&aring; O  − 
Q O
4.15 Transforming of Trigonometric
Expressions to Sum
450. &euml;&aacute;&aring; α ⋅ &euml;&aacute;&aring; β =
&Aring;&ccedil;&euml;(α − β) − &Aring;&ccedil;&euml;(α + β )
O
451. &Aring;&ccedil;&euml; α ⋅ &Aring;&ccedil;&euml; β =
&Aring;&ccedil;&euml;(α − β ) + &Aring;&ccedil;&euml;(α + β )
O
452. &euml;&aacute;&aring; α ⋅ &Aring;&ccedil;&euml; β =
&euml;&aacute;&aring;(α − β ) + &euml;&aacute;&aring;(α + β )
O
453. &iacute;~&aring; α ⋅ &iacute;~&aring; β =
&iacute;~&aring; α + &iacute;~&aring; β
&Aring;&ccedil;&iacute; α + &Aring;&ccedil;&iacute; β
454. &Aring;&ccedil;&iacute; α ⋅ &Aring;&ccedil;&iacute; β =
&Aring;&ccedil;&iacute; α + &Aring;&ccedil;&iacute; β
&iacute;~&aring; α + &iacute;~&aring; β
455. &iacute;~&aring; α ⋅ &Aring;&ccedil;&iacute; β =
&iacute;~&aring; α + &Aring;&ccedil;&iacute; β
&Aring;&ccedil;&iacute; α + &iacute;~&aring; β
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CHAPTER 4. TRIGONOMETRY
4.16 Powers of Trigonometric Functions
456. &euml;&aacute;&aring; O α =
N − &Aring;&ccedil;&euml; Oα
O
457. &euml;&aacute;&aring; P α =
P &euml;&aacute;&aring; α − &euml;&aacute;&aring; Pα
Q
458. &euml;&aacute;&aring; Q α =
&Aring;&ccedil;&euml; Qα − Q &Aring;&ccedil;&euml; Oα + P
U
459. &euml;&aacute;&aring; R α =
NM &euml;&aacute;&aring; α − R &euml;&aacute;&aring; Pα + &euml;&aacute;&aring; Rα
NS
460. &euml;&aacute;&aring; S α =
NM − NR &Aring;&ccedil;&euml; Oα + S &Aring;&ccedil;&euml; Qα − &Aring;&ccedil;&euml; Sα
PO
461. &Aring;&ccedil;&euml; O α =
N + &Aring;&ccedil;&euml; Oα
O
462. &Aring;&ccedil;&euml; P α =
P &Aring;&ccedil;&euml; α + &Aring;&ccedil;&euml; Pα
Q
463. &Aring;&ccedil;&euml; Q α =
&Aring;&ccedil;&euml; Qα + Q &Aring;&ccedil;&euml; Oα + P
U
464. &Aring;&ccedil;&euml; R α =
NM &Aring;&ccedil;&euml; α + R &euml;&aacute;&aring; Pα + &Aring;&ccedil;&euml; Rα
NS
465. &Aring;&ccedil;&euml; S α =
NM + NR &Aring;&ccedil;&euml; Oα + S &Aring;&ccedil;&euml; Qα + &Aring;&ccedil;&euml; Sα
PO
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CHAPTER 4. TRIGONOMETRY
4.17 Graphs of Inverse Trigonometric
Functions
466. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; p&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; I − N ≤ &ntilde; ≤ N I −
π
π
≤ ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; ≤ K
O
O
Figure 66.
467. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; `&ccedil;&euml;&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; I − N ≤ &ntilde; ≤ N I M ≤ ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; ≤ π K
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CHAPTER 4. TRIGONOMETRY
Figure 67.
468. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; q~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&iacute;~&aring; &ntilde; I − ∞ ≤ &ntilde; ≤ ∞ I −
π
π
&lt; ~&ecirc;&Aring;&iacute;~&aring; &ntilde; &lt; K
O
O
Figure 68.
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CHAPTER 4. TRIGONOMETRY
469. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; `&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; I − ∞ ≤ &ntilde; ≤ ∞ I M &lt; ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; &lt; π K
Figure 69.
470. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; p&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
 π  π 
&oacute; = ~&ecirc;&Aring;&euml;&Eacute;&Aring; &ntilde; I &ntilde; ∈ (− ∞I − N] ∪ [NI ∞ )I ~&ecirc;&Aring; &euml;&Eacute;&Aring; &ntilde; ∈ MI  ∪  I πK
 O  O 
Figure 70.
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471. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; `&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
 π   π
&oacute; = ~&ecirc;&Aring;&Aring;&euml;&Aring; &ntilde; I &ntilde; ∈ (− ∞I − N] ∪ [NI ∞ )I ~&ecirc;&Aring; &Aring;&euml;&Aring; &ntilde; ∈ − I M  ∪  MI K
 O   O
Figure 71.
4.18 Principal Values of Inverse
Trigonometric Functions
472.
&ntilde;
~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde;
~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde;
&ntilde;
~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde;
~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde;
M
M&deg;
VM&deg;
N
−
O
− PM&deg;
NOM&deg;
N
O
PM&deg;
SM&deg;
O
−
O
O
O
QR&deg;
QR&deg;
P
−
O
− QR&deg;
− SM&deg;
NPR&deg;
NRM&deg;
P
O
SM&deg;
PM&deg;
−N
− VM&deg;
NUM&deg;
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N
VM&deg;
M&deg;
CHAPTER 4. TRIGONOMETRY
473.
&ntilde;
M
P
P
N
P
~&ecirc;&Aring;&iacute;~&aring; &ntilde;
M&deg;
PM&deg;
QR&deg;
SM&deg;
− PM&deg;
~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde;
VM&deg;
SM&deg;
QR&deg;
PM&deg;
NOM&deg;
−
P
P
4.19 Relations between Inverse
Trigonometric Functions
474. ~&ecirc;&Aring;&euml;&aacute;&aring;(− &ntilde; ) = − ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde;
475. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; =
π
− ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde;
O
476. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring;&Aring;&ccedil;&euml; N − &ntilde; O I M ≤ &ntilde; ≤ N K
477. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = − ~&ecirc;&Aring;&Aring;&ccedil;&euml; N − &ntilde; O I − N ≤ &ntilde; ≤ M K
478. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring;&iacute;~&aring;
&ntilde;
N− &ntilde;
O
O
I &ntilde; &lt; NK
479. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute;
N− &ntilde;O
I M &lt; &ntilde; ≤ NK
&ntilde;
480. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute;
N− &ntilde;O
− π I −N≤ &ntilde; &lt; M K
&ntilde;
481. ~&ecirc;&Aring;&Aring;&ccedil;&euml;(− &ntilde; ) = π − ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde;
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−N
− QR&deg;
NPR&deg;
− P
− SM&deg;
NRM&deg;
CHAPTER 4. TRIGONOMETRY
482. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; =
π
− ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde;
O
483. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = ~&ecirc;&Aring;&euml;&aacute;&aring; N − &ntilde; O I M ≤ &ntilde; ≤ N K
484. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = π − ~&ecirc;&Aring;&euml;&aacute;&aring; N − &ntilde; O I − N ≤ &ntilde; ≤ M K
485. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = ~&ecirc;&Aring;&iacute;~&aring;
N− &ntilde;O
I M &lt; &ntilde; ≤ NK
&ntilde;
N− &ntilde;O
I −N≤ &ntilde; &lt; M K
&ntilde;
486. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = π + ~&ecirc;&Aring;&iacute;~&aring;
487. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute;
&ntilde;
N− &ntilde;O
I −N≤ &ntilde; ≤ NK
488. ~&ecirc;&Aring;&iacute;~&aring;(− &ntilde; ) = − ~&ecirc;&Aring;&iacute;~&aring; &ntilde;
489. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; =
π
− ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde;
O
490. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring;&euml;&aacute;&aring;
491. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring;&Aring;&ccedil;&euml;
492. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = − ~&ecirc;&Aring;&Aring;&ccedil;&euml;
&ntilde;
N+ &ntilde;O
N
N+ &ntilde;O
I &ntilde; ≥MK
N
N+ &ntilde;O
I &ntilde; ≤MK
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CHAPTER 4. TRIGONOMETRY
493. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; =
π
N
− ~&ecirc;&Aring;&iacute;~&aring; I &ntilde; &gt; M K
&ntilde;
O
π
N
494. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = − − ~&ecirc;&Aring;&iacute;~&aring; I &ntilde; &lt; M K
&ntilde;
O
495. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute;
N
I &ntilde; &gt;MK
&ntilde;
N
496. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute; − π I &ntilde; &lt; M K
&ntilde;
497. ~&ecirc;&Aring; &Aring;&ccedil;&iacute;(− &ntilde; ) = π − ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde;
498. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; =
π
− ~&ecirc;&Aring;&iacute;~&aring; &ntilde;
O
499. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = ~&ecirc;&Aring;&euml;&aacute;&aring;
N
N+ &ntilde;O
N
500. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = π − ~&ecirc;&Aring;&euml;&aacute;&aring;
501. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = ~&ecirc;&Aring;&Aring;&ccedil;&euml;
502. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = ~&ecirc;&Aring;&iacute;~&aring;
I &ntilde; &gt;MK
N+ &ntilde;O
I &ntilde; &lt;MK
&ntilde;
N+ &ntilde;O
N
I &ntilde; &gt; MK
&ntilde;
503. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = π + ~&ecirc;&Aring;&iacute;~&aring;
N
I &ntilde; &lt;MK
&ntilde;
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CHAPTER 4. TRIGONOMETRY
4.20 Trigonometric Equations
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
504. &euml;&aacute;&aring; &ntilde; = ~ I &ntilde; = (− N) ~&ecirc;&Aring;&euml;&aacute;&aring; ~ + π&aring;
&aring;
505. &Aring;&ccedil;&euml; &ntilde; = ~ I &ntilde; = &plusmn; ~&ecirc;&Aring;&Aring;&ccedil;&euml; ~ + Oπ&aring;
506. &iacute;~&aring; &ntilde; = ~ I &ntilde; = ~&ecirc;&Aring;&iacute;~&aring; ~ + π&aring;
507. &Aring;&ccedil;&iacute; &ntilde; = ~ I &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute; ~ + π&aring;
4.21 Relations to Hyperbolic Functions
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; &igrave;&aring;&aacute;&iacute;W &aacute;
508. &euml;&aacute;&aring;(&aacute;&ntilde; ) = &aacute; &euml;&aacute;&aring;&Uuml; &ntilde;
509. &iacute;~&aring;(&aacute;&ntilde; ) = &aacute; &iacute;~&aring;&Uuml; &ntilde;
510. &Aring;&ccedil;&iacute;(&aacute;&ntilde; ) = −&aacute; &Aring;&ccedil;&iacute;&Uuml; &ntilde;
511. &euml;&Eacute;&Aring;(&aacute;&ntilde; ) = &euml;&Eacute;&Aring;&Uuml; &ntilde;
512. &Aring;&euml;&Aring;(&aacute;&ntilde; ) = −&aacute; &Aring;&euml;&Aring;&Uuml; &ntilde;
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Chapter 5
Matrices and Determinants
j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;W ^I _I `
b&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde;W ~ &aacute; I &Auml;&aacute; I ~ &aacute;&agrave; I &Auml;&aacute;&agrave; I &Aring; &aacute;&agrave;
a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute; &ccedil;&Ntilde; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde;W &Ccedil;&Eacute;&iacute; ^
j&aacute;&aring;&ccedil;&ecirc; &ccedil;&Ntilde; ~&aring; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; ~ &aacute;&agrave; W j &aacute;&agrave;
`&ccedil;&Ntilde;~&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; ~&aring; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; ~ &aacute;&agrave; W ` &aacute;&agrave;
&uacute;
q&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute; &ccedil;&Ntilde; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde;W ^ q I ^
^&Ccedil;&agrave;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde;W ~&Ccedil;&agrave; ^
q&ecirc;~&Aring;&Eacute; &ccedil;&Ntilde; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde;W &iacute;&ecirc; ^
f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; &ccedil;&Ntilde; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde;W ^ −N
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &acirc;
o&Eacute;~&auml; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &ntilde; &aacute;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &atilde;I &aring;
5.1 Determinants
513. p&Eacute;&Aring;&ccedil;&aring;&Ccedil; l&ecirc;&Ccedil;&Eacute;&ecirc; a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;
~ &Auml;N
&Ccedil;&Eacute;&iacute; ^ = N
= ~ N &Auml; O − ~ O &Auml;N
~ O &Auml;O
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CHAPTER 5. MATRICES AND DETERMINANTS
514. q&Uuml;&aacute;&ecirc;&Ccedil; l&ecirc;&Ccedil;&Eacute;&ecirc; a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;
~NN ~NO ~NP
&Ccedil;&Eacute;&iacute; ^ = ~ ON ~ OO ~ OP = ~NN~ OO~ PP + ~NO~ OP~ PN + ~ NP~ ON~ PO −
~ PN ~ PO ~ PP
− ~NN~ OP~ PO − ~NO~ ON~ PP − ~ NP~ OO~ PN
515. p~&ecirc;&ecirc;&igrave;&euml; o&igrave;&auml;&Eacute; E^&ecirc;&ecirc;&ccedil;&iuml; o&igrave;&auml;&Eacute;F
Figure 72.
516. k-&iacute;&Uuml; l&ecirc;&Ccedil;&Eacute;&ecirc; a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;
~NN ~NO K ~N&agrave; K ~N&aring;
~ ON ~ OO K ~ O &agrave; K ~ O&aring;
K K K K K K
&Ccedil;&Eacute;&iacute; ^ =
~ &aacute;N ~ &aacute; O K ~ &aacute;&agrave; K ~ &aacute;&aring;
K K K K K K
~ &aring;N ~ &aring; O K ~ &aring;&agrave; K ~ &aring;&aring;
517. j&aacute;&aring;&ccedil;&ecirc;
q&Uuml;&Eacute; &atilde;&aacute;&aring;&ccedil;&ecirc; j &aacute;&agrave; ~&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&Eacute;&Ccedil; &iuml;&aacute;&iacute;&Uuml; &iacute;&Uuml;&Eacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; ~ &aacute;&agrave; &ccedil;&Ntilde; &aring;-&iacute;&Uuml; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;
&atilde;~&iacute;&ecirc;&aacute;&ntilde; ^ &aacute;&euml; &iacute;&Uuml;&Eacute; (&aring; − N) -&iacute;&Uuml; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;&Eacute;&Ccedil; &Ntilde;&ecirc;&ccedil;&atilde;
&iacute;&Uuml;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; ^ &Auml;&oacute; &Ccedil;&Eacute;&auml;&Eacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &aacute;&iacute;&euml; &aacute;-&iacute;&Uuml; &ecirc;&ccedil;&iuml; ~&aring;&Ccedil; &agrave;-&iacute;&Uuml; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;K
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CHAPTER 5. MATRICES AND DETERMINANTS
518. `&ccedil;&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;
&aacute; +&agrave;
` &aacute;&agrave; = (− N) j &aacute;&agrave;
519. i~&eacute;&auml;~&Aring;&Eacute; b&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &aring;-&iacute;&Uuml; l&ecirc;&Ccedil;&Eacute;&ecirc; a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;
i~&eacute;&auml;~&Aring;&Eacute; &Eacute;&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring; &Auml;&oacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &aacute;-&iacute;&Uuml; &ecirc;&ccedil;&iuml;
&aring;
&Ccedil;&Eacute;&iacute; ^ = ∑ ~ &aacute;&agrave;` &aacute;&agrave; I &aacute; = NI OI KI &aring; K
&agrave;=N
i~&eacute;&auml;~&Aring;&Eacute; &Eacute;&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring; &Auml;&oacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &agrave;-&iacute;&Uuml; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;
&aring;
&Ccedil;&Eacute;&iacute; ^ = ∑ ~ &aacute;&agrave;` &aacute;&agrave; I &agrave; = NI OI KI &aring; K
&aacute; =N
5.2 Properties of Determinants
520. q&Uuml;&Eacute; &icirc;~&auml;&igrave;&Eacute; &ccedil;&Ntilde; ~ &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute; &ecirc;&Eacute;&atilde;~&aacute;&aring;&euml; &igrave;&aring;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil; &aacute;&Ntilde; &ecirc;&ccedil;&iuml;&euml; ~&ecirc;&Eacute;
&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil; &iacute;&ccedil; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml; ~&aring;&Ccedil; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml; &iacute;&ccedil; &ecirc;&ccedil;&iuml;&euml;K
~N ~ O ~N &Auml;N
=
&Auml;N &Auml;O ~ O &Auml;O
521. f&Ntilde; &iacute;&iuml;&ccedil; &ecirc;&ccedil;&iuml;&euml; E&ccedil;&ecirc; &iacute;&iuml;&ccedil; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;F ~&ecirc;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;I &iacute;&Uuml;&Eacute; &euml;&aacute;&Ouml;&aring; &ccedil;&Ntilde;
&iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute; &aacute;&euml; &Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;K
~N &Auml;N
~ &Auml;O
=− O
~ O &Auml;O
~N &Auml;N
522. f&Ntilde; &iacute;&iuml;&ccedil; &ecirc;&ccedil;&iuml;&euml; E&ccedil;&ecirc; &iacute;&iuml;&ccedil; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;F ~&ecirc;&Eacute; &aacute;&Ccedil;&Eacute;&aring;&iacute;&aacute;&Aring;~&auml;I &iacute;&Uuml;&Eacute; &icirc;~&auml;&igrave;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute;
&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute; &aacute;&euml; &ograve;&Eacute;&ecirc;&ccedil;K
~N ~N
=M
~O ~O
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CHAPTER 5. MATRICES AND DETERMINANTS
523. f&Ntilde; &iacute;&Uuml;&Eacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; ~&aring;&oacute; &ecirc;&ccedil;&iuml; E&ccedil;&ecirc; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;F ~&ecirc;&Eacute; &atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Eacute;&Ccedil; &Auml;&oacute;
~ &Aring;&ccedil;&atilde;&atilde;&ccedil;&aring; &Ntilde;~&Aring;&iacute;&ccedil;&ecirc;I &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute; &aacute;&euml; &atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;~&iacute;
&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;K
&acirc;~ N &acirc;&Auml;N
~ &Auml;N
=&acirc; N
~ O &Auml;O
~ O &Auml;O
524. f&Ntilde; &iacute;&Uuml;&Eacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; ~&aring;&oacute; &ecirc;&ccedil;&iuml; E&ccedil;&ecirc; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;F ~&ecirc;&Eacute; &aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&Eacute;&Ccedil; E&ccedil;&ecirc;
&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&Eacute;&Ccedil;F&Auml;&oacute; &Eacute;&egrave;&igrave;~&auml; &atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ecirc;&ecirc;&Eacute;&euml;&eacute;&ccedil;&aring;&Ccedil;&aacute;&aring;&Ouml; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;
&ccedil;&Ntilde; ~&aring;&oacute; &ccedil;&iacute;&Uuml;&Eacute;&ecirc; &ecirc;&ccedil;&iuml; E&ccedil;&ecirc; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;FI &iacute;&Uuml;&Eacute; &icirc;~&auml;&igrave;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;
&aacute;&euml; &igrave;&aring;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;K
~N + &acirc;&Auml;N &Auml;N ~N &Auml;N
=
~ O + &acirc;&Auml;O &Auml;O ~ O &Auml;O
5.3 Matrices
525. a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;
^&aring; &atilde; &times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde; ^ &aacute;&euml; ~ &ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc; ~&ecirc;&ecirc;~&oacute; &ccedil;&Ntilde; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; E&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; &ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;F &iuml;&aacute;&iacute;&Uuml; &atilde; &ecirc;&ccedil;&iuml;&euml; ~&aring;&Ccedil; &aring; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;K
 ~ NN ~ NO K ~ N&aring; 
~
~ OO K ~ O&aring; 
ON


^ = ~ &aacute;&agrave; =
 M
M
M 


~ &atilde;N ~ &atilde; O K ~ &atilde;&aring; 
[ ]
526. p&egrave;&igrave;~&ecirc;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &aacute;&euml; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde; &ccedil;&Ntilde; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc; &aring; &times; &aring; K
[ ]
527. ^ &euml;&egrave;&igrave;~&ecirc;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; ~ &aacute;&agrave; &aacute;&euml; &euml;&oacute;&atilde;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; &aacute;&Ntilde; ~ &aacute;&agrave; = ~ &agrave;&aacute; I &aacute;K&Eacute;K &aacute;&iacute; &aacute;&euml;
&euml;&oacute;&atilde;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;K
[ ]
528. ^ &euml;&egrave;&igrave;~&ecirc;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; ~ &aacute;&agrave; &aacute;&euml; &euml;&acirc;&Eacute;&iuml;-&euml;&oacute;&atilde;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; &aacute;&Ntilde; ~ &aacute;&agrave; = −~ &agrave;&aacute; K
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529. a&aacute;~&Ouml;&ccedil;&aring;~&auml; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &aacute;&euml; ~ &euml;&egrave;&igrave;~&ecirc;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &iuml;&aacute;&iacute;&Uuml; ~&auml;&auml; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; &ograve;&Eacute;&ecirc;&ccedil;
&Eacute;&ntilde;&Aring;&Eacute;&eacute;&iacute; &iacute;&Uuml;&ccedil;&euml;&Eacute; &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;K
530. r&aring;&aacute;&iacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &aacute;&euml; ~ &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &aacute;&aring; &iuml;&Uuml;&aacute;&Aring;&Uuml; &iacute;&Uuml;&Eacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&aring;
&iacute;&Uuml;&Eacute; &auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml; ~&ecirc;&Eacute; ~&auml;&auml; &igrave;&aring;&aacute;&iacute;&oacute;K q&Uuml;&Eacute; &igrave;&aring;&aacute;&iacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &aacute;&euml;
&Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil; &Auml;&oacute; fK
531. ^ &aring;&igrave;&auml;&auml; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &aacute;&euml; &ccedil;&aring;&Eacute; &iuml;&Uuml;&ccedil;&euml;&Eacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; ~&ecirc;&Eacute; ~&auml;&auml; &ograve;&Eacute;&ecirc;&ccedil;K
5.4 Operations with Matrices
532. q&iuml;&ccedil; &atilde;~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml; ^ ~&aring;&Ccedil; _ ~&ecirc;&Eacute; &Eacute;&egrave;&igrave;~&auml; &aacute;&Ntilde;I ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde;I &iacute;&Uuml;&Eacute;&oacute; ~&ecirc;&Eacute; &Auml;&ccedil;&iacute;&Uuml;
&ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;~&atilde;&Eacute; &euml;&Uuml;~&eacute;&Eacute; &atilde; &times; &aring; ~&aring;&Ccedil; &Aring;&ccedil;&ecirc;&ecirc;&Eacute;&euml;&eacute;&ccedil;&aring;&Ccedil;&aacute;&aring;&Ouml; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml; ~&ecirc;&Eacute;
&Eacute;&egrave;&igrave;~&auml;K
533. q&iuml;&ccedil; &atilde;~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml; ^ ~&aring;&Ccedil; _ &Aring;~&aring; &Auml;&Eacute; ~&Ccedil;&Ccedil;&Eacute;&Ccedil; E&ccedil;&ecirc; &euml;&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&Eacute;&Ccedil;F &ccedil;&Ntilde;I ~&aring;&Ccedil;
&ccedil;&aring;&auml;&oacute; &aacute;&Ntilde;I &iacute;&Uuml;&Eacute;&oacute; &Uuml;~&icirc;&Eacute; &iacute;&Uuml;&Eacute; &euml;~&atilde;&Eacute; &euml;&Uuml;~&eacute;&Eacute; &atilde; &times; &aring; K f&Ntilde;
 ~NN ~NO K ~N&aring; 
~
~ OO K ~ O&aring; 
I
^ = ~ &aacute;&agrave; =  ON
 M
M
M 


~ &atilde;N ~ &atilde; O K ~ &atilde;&aring; 
 &Auml;NN &Auml;NO K &Auml;N&aring; 
&Auml;
&Auml;OO K &Auml;O&aring; 
I
_ = &Auml;&aacute;&agrave; =  ON
 M
M
M 


&Auml;&atilde;N &Auml;&atilde; O K &Auml;&atilde;&aring; 
[ ]
[ ]
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&iacute;&Uuml;&Eacute;&aring;
~NO + &Auml;NO K ~ N&aring; + &Auml;N&aring; 
 ~NN + &Auml;NN
~ +&Auml;
~ OO + &Auml;OO K ~ O&aring; + &Auml;O&aring; 
ON
ON

K
^+_=


M
M
M


~ &atilde;N + &Auml;&atilde;N ~ &atilde; O + &Auml;&atilde; O K ~ &atilde;&aring; + &Auml;&atilde;&aring; 
[ ]
534. f&Ntilde; &acirc; &aacute;&euml; ~ &euml;&Aring;~&auml;~&ecirc;I ~&aring;&Ccedil; ^ = ~ &aacute;&agrave; &aacute;&euml; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde;I &iacute;&Uuml;&Eacute;&aring;
 &acirc;~NN &acirc;~NO K &acirc;~N&aring; 
 &acirc;~
&acirc;~ OO K &acirc;~ O&aring; 
ON

K
&acirc;^ = &acirc;~ &aacute;&agrave; =
 M
M
M 


&acirc;~ &atilde;N &acirc;~ &atilde; O K &acirc;~ &atilde;&aring; 
[ ]
535. j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; q&iuml;&ccedil; j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;
q&iuml;&ccedil; &atilde;~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml; &Aring;~&aring; &Auml;&Eacute; &atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Eacute;&Ccedil; &iacute;&ccedil;&Ouml;&Eacute;&iacute;&Uuml;&Eacute;&ecirc; &ccedil;&aring;&auml;&oacute; &iuml;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute;
&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&ecirc;&euml;&iacute; &aacute;&euml; &Eacute;&egrave;&igrave;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde;
&ecirc;&ccedil;&iuml;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &euml;&Eacute;&Aring;&ccedil;&aring;&Ccedil;K
f&Ntilde;
 ~NN
~
^ = ~ &aacute;&agrave; =  ON
 M

~ &atilde;N
 &Auml;NN
&Auml;
_ = &Auml;&aacute;&agrave; =  ON
 M

 &Auml; &aring;N
[ ]
[ ]
~NO
~ OO
M
~&atilde;O
&Auml;NO
&Auml;OO
M
&Auml;&aring; O
K ~N&aring; 
K ~ O&aring; 
I
M 

K ~ &atilde;&aring; 
K &Auml;N&acirc; 
K &Auml;O &acirc; 
I
M 

K &Auml;&aring;&acirc; 
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&iacute;&Uuml;&Eacute;&aring;
 &Aring;NN &Aring;NO K &Aring;N&acirc; 
&Aring;
&Aring; OO K &Aring; O &acirc; 
ON

I
^_ = ` =
 M
M
M 


&Auml; &atilde;N &Aring; &atilde; O K &Aring; &atilde;&acirc; 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&aring;
&Aring; &aacute;&agrave; = ~ &aacute;N&Auml;N&agrave; + ~ &aacute; O &Auml;O &agrave; + K + ~ &aacute;&aring; &Auml;&aring;&agrave; = ∑ ~ &aacute; λ &Auml;λ &agrave;
E &aacute; = NI OI KI &atilde; X &agrave; = NI OI KI &acirc; FK
λ =N
q&Uuml;&igrave;&euml; &aacute;&Ntilde;
[ ]
~ NN
^ = ~ &aacute;&agrave; = 
~ ON
~ NO
~ OO
 &Auml;N 
~ NP 
I _ = [&Auml; &aacute; ] = &Auml; O  I


~ OP 
&Auml;P 
&iacute;&Uuml;&Eacute;&aring;
~ NO
~
^_ =  NN
~ ON ~ OO
&Auml; 
~ NP   N  ~ NN&Auml;N
⋅ &Auml; =
~ OP   O  ~ ON&Auml;N
&Auml;P 
~ NO &Auml; O
~ OO &Auml; O
~ NP &Auml;P 
K
~ OP &Auml;P 
536. q&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute; &ccedil;&Ntilde; ~ j~&iacute;&ecirc;&aacute;&ntilde;
f&Ntilde; &iacute;&Uuml;&Eacute; &ecirc;&ccedil;&iuml;&euml; ~&aring;&Ccedil; &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml; &ccedil;&Ntilde; ~ &atilde;~&iacute;&ecirc;&aacute;&ntilde; ~&ecirc;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;I &iacute;&Uuml;&Eacute;&aring;
&iacute;&Uuml;&Eacute; &aring;&Eacute;&iuml; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &aacute;&euml; &Aring;~&auml;&auml;&Eacute;&Ccedil; &iacute;&Uuml;&Eacute; &iacute;&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;~&auml; &atilde;~&iacute;&ecirc;&aacute;&ntilde;K
f&Ntilde; ^ &aacute;&euml; &iacute;&Uuml;&Eacute; &ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;~&auml; &atilde;~&iacute;&ecirc;&aacute;&ntilde;I &aacute;&iacute;&euml; &iacute;&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute; &aacute;&euml; &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil; ^ q &ccedil;&ecirc;
&uacute;
^K
537. q&Uuml;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; ^ &aacute;&euml; &ccedil;&ecirc;&iacute;&Uuml;&ccedil;&Ouml;&ccedil;&aring;~&auml; &aacute;&Ntilde; ^^ q = f K
538. f&Ntilde; &iacute;&Uuml;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; ^_ &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;I &iacute;&Uuml;&Eacute;&aring;
(^_ )q = _ q ^ q K
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539. ^&Ccedil;&agrave;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; j~&iacute;&ecirc;&aacute;&ntilde;
f&Ntilde; ^ &aacute;&euml; ~ &euml;&egrave;&igrave;~&ecirc;&Eacute; &aring; &times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde;I &aacute;&iacute;&euml; ~&Ccedil;&agrave;&ccedil;&aacute;&aring;&iacute;I &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil; &Auml;&oacute; ~&Ccedil;&agrave; ^ I
&aacute;&euml; &iacute;&Uuml;&Eacute; &iacute;&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &ccedil;&Ntilde; &Aring;&ccedil;&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;&euml; ` &aacute;&agrave; &ccedil;&Ntilde; ^W
[ ]
~&Ccedil;&agrave; ^ = ` &aacute;&agrave; K
q
540. q&ecirc;~&Aring;&Eacute; &ccedil;&Ntilde; ~ j~&iacute;&ecirc;&aacute;&ntilde;
f&Ntilde; ^ &aacute;&euml; ~ &euml;&egrave;&igrave;~&ecirc;&Eacute; &aring; &times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde;I &aacute;&iacute;&euml; &iacute;&ecirc;~&Aring;&Eacute;I &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&ecirc; ^ I &aacute;&euml;
&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; &iacute;&ccedil; &Auml;&Eacute; &iacute;&Uuml;&Eacute; &euml;&igrave;&atilde; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &iacute;&Eacute;&ecirc;&atilde;&euml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml; &Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;W
&iacute;&ecirc; ^ = ~NN + ~ OO + K + ~ &aring;&aring; K
541. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; &ccedil;&Ntilde; ~ j~&iacute;&ecirc;&aacute;&ntilde;
f&Ntilde; ^ &aacute;&euml; ~ &euml;&egrave;&igrave;~&ecirc;&Eacute; &aring; &times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &iuml;&aacute;&iacute;&Uuml; ~ &aring;&ccedil;&aring;&euml;&aacute;&aring;&Ouml;&igrave;&auml;~&ecirc; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;
&Ccedil;&Eacute;&iacute; ^ I &iacute;&Uuml;&Eacute;&aring; &aacute;&iacute;&euml; &aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; ^ −N &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
~&Ccedil;&agrave; ^
K
^ −N =
&Ccedil;&Eacute;&iacute; ^
542. f&Ntilde; &iacute;&Uuml;&Eacute; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; ^_ &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;I &iacute;&Uuml;&Eacute;&aring;
(^_)−N = _ −N^ −N K
543. f&Ntilde; ^ &aacute;&euml; ~ &euml;&egrave;&igrave;~&ecirc;&Eacute; &aring; &times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde;I &iacute;&Uuml;&Eacute; &Eacute;&aacute;&Ouml;&Eacute;&aring;&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; u &euml;~&iacute;&aacute;&euml;&Ntilde;&oacute;
&iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
^u = λu I
&iuml;&Uuml;&aacute;&auml;&Eacute; &iacute;&Uuml;&Eacute; &Eacute;&aacute;&Ouml;&Eacute;&aring;&icirc;~&auml;&igrave;&Eacute;&euml; λ &euml;~&iacute;&aacute;&euml;&Ntilde;&oacute; &iacute;&Uuml;&Eacute; &Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
^ − λf = M K
5.5 Systems of Linear Equations
s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &ntilde;I &oacute;I &ograve;I &ntilde; N I &ntilde; O I K
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ~ N I ~ O I ~ P I &Auml;N I ~ NN I ~ NO I K
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a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;W aI a&ntilde; I a&oacute; I a&ograve;
j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;W ^I _I u
~ &ntilde; + &Auml;N&oacute; = &Ccedil;N
I
544.  N
~ O &ntilde; + &Auml;O &oacute; = &Ccedil; O
a&oacute;
a
E`&ecirc;~&atilde;&Eacute;&ecirc;∞&euml; &ecirc;&igrave;&auml;&Eacute;FI
&ntilde;= &ntilde; I &oacute;=
a
a
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
~ &Auml;N
a= N
= ~N&Auml;O − ~ O &Auml;N I
~ O &Auml;O
&Ccedil; &Auml;N
a&ntilde; = N
= &Ccedil;N&Auml;O − &Ccedil; O &Auml;N I
&Ccedil; O &Auml;O
~ &Ccedil;N
a&oacute; = N
= ~N&Ccedil; O − ~ O&Ccedil;N K
~ O &Ccedil;O
545. f&Ntilde; a ≠ M I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&oacute;&euml;&iacute;&Eacute;&atilde; &Uuml;~&euml; ~ &euml;&aacute;&aring;&Ouml;&auml;&Eacute; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;W
a&oacute;
a
K
&ntilde;= &ntilde; I &oacute;=
a
a
f&Ntilde; a = M ~&aring;&Ccedil; a&ntilde; ≠ M E&ccedil;&ecirc; a&oacute; ≠ M FI &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&oacute;&euml;&iacute;&Eacute;&atilde; &Uuml;~&euml; &aring;&ccedil;
&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;K
f&Ntilde; a = a&ntilde; = a&oacute; = M I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&oacute;&euml;&iacute;&Eacute;&atilde; &Uuml;~&euml; &aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;&auml;&oacute; &atilde;~&aring;&oacute;
&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;K
~N&ntilde; + &Auml;N&oacute; + &Aring;N&ograve; = &Ccedil;N

546. ~ O &ntilde; + &Auml;O &oacute; + &Aring; O&ograve; = &Ccedil; O I
~ &ntilde; + &Auml; &oacute; + &Aring; &ograve; = &Ccedil;
P
P
P
 P
&ntilde;=
a&oacute;
a&ntilde;
a
I &oacute;=
I &ograve; = &ograve; E`&ecirc;~&atilde;&Eacute;&ecirc;∞&euml; &ecirc;&igrave;&auml;&Eacute;FI
a
a
a
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&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
~N
a = ~O
~P
&Auml;N
&Aring;N
&Ccedil;N
&Auml;N
&Aring;N
&Auml;O
&Aring; O I a&ntilde; = &Ccedil; O
&Auml;O
&Aring;O I
&Auml;P
&Aring;P
&Auml;P
&Aring;P
&Ccedil;P
~N
&Ccedil;N
&Aring;N
~N
&Auml;N
&Ccedil;N
a&oacute; = ~ O
~P
&Ccedil;O
&Ccedil;P
&Aring; O I a&ograve; = ~ O
&Aring;P
~P
&Auml;O
&Auml;P
&Ccedil;O K
&Ccedil;P
547. f&Ntilde; a ≠ M I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&oacute;&euml;&iacute;&Eacute;&atilde; &Uuml;~&euml; ~ &euml;&aacute;&aring;&Ouml;&auml;&Eacute; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;W
a&oacute;
a
a
I &ograve;= &ograve; K
&ntilde;= &ntilde; I &oacute;=
a
a
a
f&Ntilde; a = M ~&aring;&Ccedil; a&ntilde; ≠ M E&ccedil;&ecirc; a&oacute; ≠ M &ccedil;&ecirc; a&ograve; ≠ M FI &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&oacute;&euml;&iacute;&Eacute;&atilde;
&Uuml;~&euml; &aring;&ccedil; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;K
f&Ntilde; a = a&ntilde; = a&oacute; = a&ograve; = M I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&oacute;&euml;&iacute;&Eacute;&atilde; &Uuml;~&euml; &aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;&auml;&oacute;
&atilde;~&aring;&oacute; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;K
548. j~&iacute;&ecirc;&aacute;&ntilde; c&ccedil;&ecirc;&atilde; &ccedil;&Ntilde; ~ p&oacute;&euml;&iacute;&Eacute;&atilde; &ccedil;&Ntilde; &aring; i&aacute;&aring;&Eacute;~&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &aacute;&aring;
&aring; r&aring;&acirc;&aring;&ccedil;&iuml;&aring;&euml;
q&Uuml;&Eacute; &euml;&Eacute;&iacute; &ccedil;&Ntilde; &auml;&aacute;&aring;&Eacute;~&ecirc; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;
~NN&ntilde; N + ~ NO &ntilde; O + K + ~N&aring; &ntilde; &aring; = &Auml;N
~ &ntilde; + ~ &ntilde; + K + ~ &ntilde; = &Auml;
 ON N OO O
O&aring; &aring;
O

K
K
K
K
K
K
K
K
K
K
K
K

~ &aring;N&ntilde; N + ~ &aring; O &ntilde; O + K + ~ &aring;&aring; &ntilde; &aring; = &Auml;&aring;
&Aring;~&aring; &Auml;&Eacute; &iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring; &aacute;&aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde; &Ntilde;&ccedil;&ecirc;&atilde;
 ~ NN ~ NO K ~ N&aring;   &ntilde; N   &Auml;N 

    
 ~ ON ~ OO K ~ O&aring;   &ntilde; O   &Auml; O 
⋅
=
I
 M
M
M   M   M 

    
~
    
 &aring;N ~ &aring; O K ~ &aring;&aring;   &ntilde; &aring;   &Auml; &aring; 
&aacute;K&Eacute;K
^⋅u = _I
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CHAPTER 5. MATRICES AND DETERMINANTS
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
 ~ NN

~
^ =  ON
M

~
 &aring;N
~ NO K ~ N&aring; 
 &ntilde;N 
 &Auml;N 
 
 

~ OO K ~ O&aring; 
 &ntilde;O 
 &Auml;O 
=
=
u
_
I
I
 M 
 M K
M
M 
 
 

&ntilde; 
&Auml; 
~ &aring; O K ~ &aring;&aring; 
 &aring;
 &aring;
549. p&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ p&Eacute;&iacute; &ccedil;&Ntilde; i&aacute;&aring;&Eacute;~&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &aring; &times; &aring;
u = ^ −N ⋅ _ I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ^ −N &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; &ccedil;&Ntilde; ^K
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Chapter 6
Vectors
r r r r →
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W &igrave; I &icirc; I &iuml; I &ecirc; I ^_ I &pound;
r r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W &igrave; I &icirc; I &pound;
r r r
r&aring;&aacute;&iacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W &aacute; I &agrave; I &acirc;
r
k&igrave;&auml;&auml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;W M
r
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &igrave; W uN I vN I wN
r
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &icirc; W u O I vO I wO
p&Aring;~&auml;~&ecirc;&euml;W λ I &micro;
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;W &Aring;&ccedil;&euml; α I &Aring;&ccedil;&euml; β I &Aring;&ccedil;&euml; γ
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&iuml;&ccedil; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W θ
6.1 Vector Coordinates
550. r&aring;&aacute;&iacute; s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;
r
&aacute; = (NI MI M) I
r
&agrave; = (MI NI M) I
r
&acirc; = (MI MI N) I
r r r
&aacute; = &agrave; = &acirc; = NK
r
r
r
r →
551. &ecirc; = ^_ = (&ntilde; N − &ntilde; M ) &aacute; + (&oacute; N − &oacute; M ) &agrave; + (&ograve; N − &ograve; M ) &acirc;
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CHAPTER 6. VECTORS
Figure 73.
552.
→
r
&ecirc; = ^_ =
(&ntilde; N − &ntilde; M )O + (&oacute;N − &oacute; M )O + (&ograve;N − &ograve; M )O
→
→
r
r
553. f&Ntilde; ^_ = &ecirc; I &iacute;&Uuml;&Eacute;&aring; _^ = − &ecirc; K
Figure 74.
r
554. u = &ecirc; &Aring;&ccedil;&euml; α I
r
v = &ecirc; &Aring;&ccedil;&euml; β I
r
w = &ecirc; &Aring;&ccedil;&euml; γ K
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CHAPTER 6. VECTORS
Figure 75.
r
r
555. f&Ntilde; &ecirc; (uI v I w ) = &ecirc;N (uN I vN I wN ) I &iacute;&Uuml;&Eacute;&aring;
u = uN I v = vN I w = wN K
r r r
556. &iuml; = &igrave; + &icirc;
Figure 76.
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CHAPTER 6. VECTORS
Figure 77.
r
r r r r
557. &iuml; = &igrave;N + &igrave; O + &igrave;P + K + &igrave; &aring;
Figure 78.
558. `&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute; i~&iuml;
r r r r
&igrave;+ &icirc; =&icirc;+&igrave;
559. ^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute; i~&iuml;
(&igrave;r + &icirc;r ) + &iuml;r = &igrave;r + (&icirc;r + &iuml;r )
r r
560. &igrave; + &icirc; = (uN + u O I vN + vO I wN + wO )
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CHAPTER 6. VECTORS
6.3 Vector Subtraction
r r r r r r
561. &iuml; = &igrave; − &icirc; &aacute;&Ntilde; &icirc; + &iuml; = &igrave; K
Figure 79.
Figure 80.
r r r
r
562. &igrave; − &icirc; = &igrave; + (− &icirc; )
r r r
563. &igrave; − &igrave; = M = (MI MI M )
564.
r
M =M
r r
565. &igrave; − &icirc; = (uN − u O I vN − vO I wN − w O ) I
6.4 Scaling Vectors
r
r
566. &iuml; = λ&igrave;
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CHAPTER 6. VECTORS
Figure 81.
567.
r
r
&iuml; = λ⋅&igrave;
r
568. λ&igrave; = (λuI λv I λw )
r r
569. λ&igrave; = &igrave;λ
570.
(λ + &micro; ) &igrave;r = λ&igrave;r + &micro;&igrave;r
r
r
r
571. λ(&micro;&igrave; ) = &micro;(λ&igrave; ) = (λ&micro; )&igrave;
r r
r
r
572. λ(&igrave; + &icirc; ) = λ&igrave; + λ&icirc;
6.5 Scalar Product
r
r
573. p&Aring;~&auml;~&ecirc; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; &ccedil;&Ntilde; s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &igrave; ~&aring;&Ccedil; &icirc;
r r r r
&igrave; ⋅ &icirc; = &igrave; ⋅ &icirc; ⋅ &Aring;&ccedil;&euml; θ I
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; θ &aacute;&euml; &iacute;&Uuml;&Eacute; ~&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &igrave; ~&aring;&Ccedil; &icirc; K
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CHAPTER 6. VECTORS
Figure 82.
574. p&Aring;~&auml;~&ecirc; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; &aacute;&aring; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; c&ccedil;&ecirc;&atilde;
r
r
f&Ntilde; &igrave; = (uN I vN I wN ) I &icirc; = (u O I vO I w O ) I &iacute;&Uuml;&Eacute;&aring;
r r
&igrave; ⋅ &icirc; = uNu O + vNvO + wNwO K
575. ^&aring;&Ouml;&auml;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;
r
r
f&Ntilde; &igrave; = (uN I vN I wN ) I &icirc; = (u O I vO I w O ) I &iacute;&Uuml;&Eacute;&aring;
uNu O + vNvO + wNw O
K
&Aring;&ccedil;&euml; θ =
O
O
O
O
O
O
uN + vN + wN u O + vO + w O
576. `&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute; m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;
r r r r
&igrave;⋅&icirc; = &icirc; ⋅&igrave;
577. ^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute; m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;
(λ&igrave;r ) ⋅ (&micro;&icirc;r ) = λ&micro;&igrave;r ⋅ &icirc;r
578. a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&Eacute; m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;
r r r r r r r
&igrave; ⋅ (&icirc; + &iuml; ) = &igrave; ⋅ &icirc; + &igrave; ⋅ &iuml;
π
r r
r r
579. &igrave; ⋅ &icirc; = M &aacute;&Ntilde; &igrave; I &icirc; ~&ecirc;&Eacute; &ccedil;&ecirc;&iacute;&Uuml;&ccedil;&Ouml;&ccedil;&aring;~&auml; E θ = FK
O
π
r r
580. &igrave; ⋅ &icirc; &gt; M &aacute;&Ntilde; M &lt; θ &lt; K
O
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CHAPTER 6. VECTORS
π
r r
581. &igrave; ⋅ &icirc; &lt; M &aacute;&Ntilde; &lt; θ &lt; π K
O
r r r r
582. &igrave; ⋅ &icirc; ≤ &igrave; ⋅ &icirc;
r r r r
r r
583. &igrave; ⋅ &icirc; = &igrave; ⋅ &icirc; &aacute;&Ntilde; &igrave; I &icirc; ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; E θ = M FK
r
584. f&Ntilde; &igrave; = (uN I vN I wN ) I &iacute;&Uuml;&Eacute;&aring;
r r r
rO
O
O
O
&igrave; ⋅ &igrave; = &igrave; O = &igrave; = uN + vN + wN K
r r r r r r
585. &aacute; ⋅ &aacute; = &agrave; ⋅ &agrave; = &acirc; ⋅ &acirc; = N
r r r r r r
586. &aacute; ⋅ &agrave; = &agrave; ⋅ &acirc; = &acirc; ⋅ &aacute; = M
6.6 Vector Product
r
r
587. s&Eacute;&Aring;&iacute;&ccedil;&ecirc; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; &ccedil;&Ntilde; s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &igrave; ~&aring;&Ccedil; &icirc;
r r r
&igrave; &times; &icirc; = &iuml; I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
π
r r r
•
&iuml; = &igrave; ⋅ &icirc; ⋅ &euml;&aacute;&aring; θ I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; M ≤ θ ≤ X
O
r r
r r
•
&iuml; ⊥&igrave;
~&aring;&Ccedil; &iuml; ⊥ &icirc; X
r r r
•
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &igrave; I &icirc; I &iuml; &Ntilde;&ccedil;&ecirc;&atilde; ~ &ecirc;&aacute;&Ouml;&Uuml;&iacute;-&Uuml;~&aring;&Ccedil;&Eacute;&Ccedil; &euml;&Aring;&ecirc;&Eacute;&iuml;K
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CHAPTER 6. VECTORS
Figure 83.
r
&acirc;
wN
wO
r
&aacute;
r r r
588. &iuml; = &igrave; &times; &icirc; = u N
uO
r
&agrave;
vN
vO
r r r v
589. &iuml; = &igrave; &times; &icirc; =  N
 vO
wN
u
I− N
wO
uO
wN uN
I
wO uO
vN
vO
r r r r
590. p = &igrave; &times; &icirc; = &igrave; ⋅ &icirc; ⋅ &euml;&aacute;&aring; θ Ec&aacute;&Ouml;KUPF
591. ^&aring;&Ouml;&auml;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; Ec&aacute;&Ouml;KUPF
r r
&igrave;&times; &icirc;
&euml;&aacute;&aring; θ = r r
&igrave;⋅&icirc;
592. k&ccedil;&aring;&Aring;&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute; m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;
r r
r r
&igrave; &times; &icirc; = − (&icirc; &times; &igrave; )
593. ^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute; m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;
(λ&igrave;r )&times; (&micro;&icirc;r ) = λ&micro;&igrave;r &times; &icirc;r
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



CHAPTER 6. VECTORS
594. a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&Eacute; m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;
r r r r r r r
&igrave; &times; (&icirc; + &iuml; ) = &igrave; &times; &icirc; + &igrave; &times; &iuml;
r r r r
r
595. &igrave; &times; &icirc; = M &aacute;&Ntilde; &igrave; ~&aring;&Ccedil; &icirc; ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; E θ = M FK
r r r r r r r
596. &aacute; &times; &aacute; = &agrave; &times; &agrave; = &acirc; &times; &acirc; = M
r r r r r r r r r
597. &aacute; &times; &agrave; = &acirc; I &agrave; &times; &acirc; = &aacute; I &acirc; &times; &aacute; = &agrave;
6.7 Triple Product
598. p&Aring;~&auml;~&ecirc; q&ecirc;&aacute;&eacute;&auml;&Eacute; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;
[&igrave;r &icirc;r&iuml;r ] = &igrave;r ⋅ (&icirc;r &times; &iuml;r ) = &icirc;r ⋅ (&iuml;r &times; &igrave;r ) = &iuml;r ⋅ (&igrave;r &times; &icirc;r )
599.
[&igrave;r &icirc;r&iuml;r ] = [&iuml;r &igrave;r &icirc;r ] = [&icirc;r&iuml;r &igrave;r ] = −[&icirc;r&igrave;r &iuml;r ] = −[&iuml;r &icirc;r&igrave;r ] = −[&igrave;r &iuml;r &icirc;r ]
r r r
rr r
600. &acirc;&igrave; ⋅ (&icirc; &times; &iuml; ) = &acirc;[&igrave;&icirc;&iuml; ]
601. p&Aring;~&auml;~&ecirc; q&ecirc;&aacute;&eacute;&auml;&Eacute; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; &aacute;&aring; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; c&ccedil;&ecirc;&atilde;
uN vN wN
r r r
&igrave; ⋅ (&icirc; &times; &iuml; ) = u O vO w O I
uP vP wP
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
r
r
r
&igrave; = (uN I vN I wN ) I &icirc; = (u O I vO I w O ) I &iuml; = (uP I vP I wP ) K
602. s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; m~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;
r r r
s = &igrave; ⋅ (&icirc; &times; &iuml; )
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CHAPTER 6. VECTORS
Figure 84.
603. s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; m&oacute;&ecirc;~&atilde;&aacute;&Ccedil;
Nr r r
s = &igrave; ⋅ (&icirc; &times; &iuml; )
S
Figure 85.
r r r
r r
r
604. f&Ntilde; &igrave; ⋅ (&icirc; &times; &iuml; ) = M I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &igrave; I &icirc; I ~&aring;&Ccedil; &iuml; ~&ecirc;&Eacute; &auml;&aacute;&aring;&Eacute;~&ecirc;&auml;&oacute;
r
r
r
&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; I &euml;&ccedil; &iuml; = λ&igrave; + &micro;&icirc; &Ntilde;&ccedil;&ecirc; &euml;&ccedil;&atilde;&Eacute; &euml;&Aring;~&auml;~&ecirc;&euml; λ ~&aring;&Ccedil; &micro; K
r r r
r r
r
605. f&Ntilde; &igrave; ⋅ (&icirc; &times; &iuml; ) ≠ M I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &igrave; I &icirc; I ~&aring;&Ccedil; &iuml; ~&ecirc;&Eacute; &auml;&aacute;&aring;&Eacute;~&ecirc;&auml;&oacute;
&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;K
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CHAPTER 6. VECTORS
606. s&Eacute;&Aring;&iacute;&ccedil;&ecirc; q&ecirc;&aacute;&eacute;&auml;&Eacute; m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;
r r r
r r r r r r
&igrave; &times; (&icirc; &times; &iuml; ) = (&igrave; ⋅ &iuml; )&icirc; − (&igrave; ⋅ &icirc; )&iuml;
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Chapter 7
Analytic Geometry
7.1 One-Dimensional Coordinate System
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ntilde; M I &ntilde; N I &ntilde; O I &oacute; M I &oacute; N I &oacute; O
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W λ
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&iuml;&ccedil; &eacute;&ccedil;&aacute;&aring;&iacute;&euml;W &Ccedil;
607. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; m&ccedil;&aacute;&aring;&iacute;&euml;
&Ccedil; = ^_ = &ntilde; O − &ntilde; N = &ntilde; N − &ntilde; O
Figure 86.
608. a&aacute;&icirc;&aacute;&Ccedil;&aacute;&aring;&Ouml; ~ i&aacute;&aring;&Eacute; p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute; &aacute;&aring; &iacute;&Uuml;&Eacute; o~&iacute;&aacute;&ccedil; λ
&ntilde; + λ&ntilde; O
^`
I λ=
&ntilde;M = N
I λ ≠ −N K
N+ λ
`_
Figure 87.
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CHAPTER 7. ANALYTIC GEOMETRY
609. j&aacute;&Ccedil;&eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; ~ i&aacute;&aring;&Eacute; p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;
&ntilde; + &ntilde;O
I λ =NK
&ntilde;M = N
O
7.2 Two-Dimensional Coordinate System
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ntilde; M I &ntilde; N I &ntilde; O I &oacute; M I &oacute; N I &oacute; O
m&ccedil;&auml;~&ecirc; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ecirc;I ϕ
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W λ
m&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &ecirc;&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ~I &Auml;I &Aring;I
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&iuml;&ccedil; &eacute;&ccedil;&aacute;&aring;&iacute;&euml;W &Ccedil;
^&ecirc;&Eacute;~W p
610. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; m&ccedil;&aacute;&aring;&iacute;&euml;
&Ccedil; = ^_ =
(&ntilde; O − &ntilde; N )O + (&oacute; O − &oacute;N )O
Figure 88.
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CHAPTER 7. ANALYTIC GEOMETRY
611. a&aacute;&icirc;&aacute;&Ccedil;&aacute;&aring;&Ouml; ~ i&aacute;&aring;&Eacute; p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute; &aacute;&aring; &iacute;&Uuml;&Eacute; o~&iacute;&aacute;&ccedil; λ
&ntilde; + λ&ntilde; O
&oacute; + λ&oacute; O
&ntilde;M = N
I &oacute;M = N
I
N+ λ
N+ λ
^`
λ=
I λ ≠ −N K
`_
Figure 89.
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 90.
612. j&aacute;&Ccedil;&eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; ~ i&aacute;&aring;&Eacute; p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;
&ntilde; + &ntilde;O
&oacute; + &oacute;O
I &oacute;M = N
I λ =NK
&ntilde;M = N
O
O
613. `&Eacute;&aring;&iacute;&ecirc;&ccedil;&aacute;&Ccedil; Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; j&Eacute;&Ccedil;&aacute;~&aring;&euml;F &ccedil;&Ntilde; ~ q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;
&ntilde; + &ntilde; O + &ntilde;P
&oacute; + &oacute;O + &oacute;P
&ntilde;M = N
I &oacute;M = N
I
P
P
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ^(&ntilde; N I &oacute; N ) I _(&ntilde; O I &oacute; O ) I ~&aring;&Ccedil; `(&ntilde; P I &oacute; P ) ~&ecirc;&Eacute; &icirc;&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml; &ccedil;&Ntilde;
&iacute;&Uuml;&Eacute; &iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; ^_` K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 91.
614. f&aring;&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ^&aring;&Ouml;&auml;&Eacute; _&aacute;&euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;F &ccedil;&Ntilde; ~ q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;
~&ntilde; + &Auml;&ntilde; O + &Aring;&ntilde; P
~&oacute; + &Auml;&oacute; O + &Aring;&oacute; P
&ntilde;M = N
I &oacute;M = N
I
~ +&Auml;+&Aring;
~ +&Auml;+&Aring;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ~ = _` I &Auml; = `^ I &Aring; = ^_ K
Figure 92.
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CHAPTER 7. ANALYTIC GEOMETRY
615. `&aacute;&ecirc;&Aring;&igrave;&atilde;&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; p&aacute;&Ccedil;&Eacute; m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;
_&aacute;&euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;F &ccedil;&Ntilde; ~ q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;
&ntilde; NO + &oacute; NO &oacute; N N
&ntilde; N &ntilde; NO + &oacute; NO N
&ntilde;M =
&ntilde; OO + &oacute; OO
&ntilde; PO + &oacute; PO
&ntilde;N
O &ntilde;O
&ntilde;P
&oacute;O N
&oacute;P N
&oacute;N N
&oacute;O N
&oacute;P N
I &oacute;M =
&ntilde;O
&ntilde;P
&ntilde; OO + &oacute; OO N
&ntilde; PO + &oacute; PO N
&ntilde;N
O &ntilde;O
&ntilde;P
&oacute;N N
&oacute;O N
&oacute;P N
Figure 93.
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CHAPTER 7. ANALYTIC GEOMETRY
616. l&ecirc;&iacute;&Uuml;&ccedil;&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;&euml;F &ccedil;&Ntilde; ~ q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;
&oacute; N &ntilde; O &ntilde; P + &oacute; NO N
&ntilde; NO + &oacute; O &oacute; P &ntilde; N N
&ntilde;M =
&oacute;O
&oacute;P
&ntilde; P &ntilde; N + &oacute; OO N
&ntilde; N&ntilde; O + &oacute; PO N
&ntilde;N
&ntilde;O
&ntilde;P
&oacute;N N
&oacute;O N
&oacute;P N
I &oacute;M =
&ntilde; OO + &oacute; P &oacute; N
&ntilde; PO + &oacute; N&oacute; O
&ntilde;N
&ntilde;O
&ntilde;P
&ntilde;O N
&ntilde;P N
&oacute;N N
&oacute;O N
&oacute;P N
Figure 94.
617. ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;
&ntilde; N &oacute;N N
N
N &ntilde; O − &ntilde;N
p = (&plusmn; ) &ntilde; O &oacute; O N = (&plusmn; )
O
O &ntilde; P − &ntilde;N
&ntilde;P &oacute;P N
&oacute; O − &oacute;N
&oacute; P − &oacute;N
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CHAPTER 7. ANALYTIC GEOMETRY
618. ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;
N
p = (&plusmn; ) [(&ntilde; N − &ntilde; O )(&oacute; N + &oacute; O ) + (&ntilde; O − &ntilde; P )(&oacute; O + &oacute; P ) +
O
+ (&ntilde; P − &ntilde; Q )(&oacute; P + &oacute; Q ) + (&ntilde; Q − &ntilde; N )(&oacute; Q + &oacute; N )]
Figure 95.
k&ccedil;&iacute;&Eacute;W f&aring; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; SNTI SNU &iuml;&Eacute; &Aring;&Uuml;&ccedil;&ccedil;&euml;&Eacute; &iacute;&Uuml;&Eacute; &euml;&aacute;&Ouml;&aring; EHF &ccedil;&ecirc; E&yen;F &euml;&ccedil;
&iacute;&Uuml;~&iacute; &iacute;&ccedil; &Ouml;&Eacute;&iacute; ~ &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; ~&aring;&euml;&iuml;&Eacute;&ecirc; &Ntilde;&ccedil;&ecirc; ~&ecirc;&Eacute;~K
619. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; m&ccedil;&aacute;&aring;&iacute;&euml; &aacute;&aring; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
&Ccedil; = ^_ = &ecirc;NO + &ecirc;OO − O&ecirc;N&ecirc;O &Aring;&ccedil;&euml;(ϕ O − ϕN )
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 96.
620. `&ccedil;&aring;&icirc;&Eacute;&ecirc;&iacute;&aacute;&aring;&Ouml; o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &iacute;&ccedil; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
&ntilde; = &ecirc; &Aring;&ccedil;&euml; ϕ I &oacute; = &ecirc; &euml;&aacute;&aring; ϕ K
Figure 97.
621. `&ccedil;&aring;&icirc;&Eacute;&ecirc;&iacute;&aacute;&aring;&Ouml; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &iacute;&ccedil; o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
&oacute;
&ecirc; = &ntilde; O + &oacute; O I &iacute;~&aring; ϕ = K
&ntilde;
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CHAPTER 7. ANALYTIC GEOMETRY
7.3 Straight Line in Plane
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W uI vI &ntilde;I &ntilde; M I &ntilde; N I &oacute; M I &oacute; N I ~N I ~ O I &pound;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &acirc;I ~I &Auml;I &eacute;I &iacute;I ^I _I `I ^N I ^ O I &pound;
^&aring;&Ouml;&auml;&Eacute;&euml;W α I β
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml;W ϕ
r
k&ccedil;&ecirc;&atilde;~&auml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;W &aring;
r r r
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W &ecirc; I ~ I &Auml;
622. d&Eacute;&aring;&Eacute;&ecirc;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute;
^&ntilde; + _&oacute; + ` = M
623. k&ccedil;&ecirc;&atilde;~&auml; s&Eacute;&Aring;&iacute;&ccedil;&ecirc; &iacute;&ccedil; ~ p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute;
r
q&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &aring;(^I _ ) &aacute;&euml; &aring;&ccedil;&ecirc;&atilde;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; ^&ntilde; + _&oacute; + ` = M K
Figure 98.
624. b&ntilde;&eacute;&auml;&aacute;&Aring;&aacute;&iacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute; Ep&auml;&ccedil;&eacute;&Eacute;-f&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute; c&ccedil;&ecirc;&atilde;F
&oacute; = &acirc;&ntilde; + &Auml; K
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CHAPTER 7. ANALYTIC GEOMETRY
q&Uuml;&Eacute; &Ouml;&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&euml; &acirc; = &iacute;~&aring; α K
Figure 99.
625. d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute; &ccedil;&Ntilde; ~ i&aacute;&aring;&Eacute;
&oacute; − &oacute;N
&acirc; = &iacute;~&aring; α = O
&ntilde; O − &ntilde;N
Figure 100.
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CHAPTER 7. ANALYTIC GEOMETRY
626. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ i&aacute;&aring;&Eacute; d&aacute;&icirc;&Eacute;&aring; ~ m&ccedil;&aacute;&aring;&iacute; ~&aring;&Ccedil; &iacute;&Uuml;&Eacute; d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;
&oacute; = &oacute; M + &acirc; (&ntilde; − &ntilde; M ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &acirc; &aacute;&euml; &iacute;&Uuml;&Eacute; &Ouml;&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;I m(&ntilde; M I &oacute; M ) &aacute;&euml; ~ &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;K
Figure 101.
627. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ i&aacute;&aring;&Eacute; q&Uuml;~&iacute; m~&euml;&euml;&Eacute;&euml; q&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml; q&iuml;&ccedil; m&ccedil;&aacute;&aring;&iacute;&euml;
&oacute; − &oacute;N
&ntilde; − &ntilde;N
=
&oacute; O − &oacute;N &ntilde; O − &ntilde;N
&ccedil;&ecirc;
&ntilde; &oacute; N
&ntilde;N &oacute;N N = M K
&ntilde;O &oacute;O N
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 102.
628. f&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute; c&ccedil;&ecirc;&atilde;
&ntilde; &oacute;
+ =N
~ &Auml;
Figure 103.
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CHAPTER 7. ANALYTIC GEOMETRY
629. k&ccedil;&ecirc;&atilde;~&auml; c&ccedil;&ecirc;&atilde;
&ntilde; &Aring;&ccedil;&euml; β + &oacute; &euml;&aacute;&aring; β − &eacute; = M
Figure 104.
630. m&ccedil;&aacute;&aring;&iacute; a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; c&ccedil;&ecirc;&atilde;
&ntilde; − &ntilde; N &oacute; − &oacute;N
=
I
u
v
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; (uI v ) &aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; ~&aring;&Ccedil; mN (&ntilde; N I &oacute; N ) &auml;&aacute;&Eacute;&euml;
&ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 105.
631. s&Eacute;&ecirc;&iacute;&aacute;&Aring;~&auml; i&aacute;&aring;&Eacute;
&ntilde; =~
632. e&ccedil;&ecirc;&aacute;&ograve;&ccedil;&aring;&iacute;~&auml; i&aacute;&aring;&Eacute;
&oacute;=&Auml;
633. s&Eacute;&Aring;&iacute;&ccedil;&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute;
r r r
&ecirc; = ~ + &iacute;&Auml; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
l &aacute;&euml; &iacute;&Uuml;&Eacute; &ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;I
u &aacute;&euml; ~&aring;&oacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I
r
~ &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; ~ &acirc;&aring;&ccedil;&iuml;&aring; &eacute;&ccedil;&aacute;&aring;&iacute; ^ &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; I
r
&Auml; &aacute;&euml; ~ &acirc;&aring;&ccedil;&iuml;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;I &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I
&iacute; &aacute;&euml; ~ &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;I
r →
&ecirc; = lu &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; ~&aring;&oacute; &eacute;&ccedil;&aacute;&aring;&iacute; u &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 106.
634. p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute; &aacute;&aring; m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&ccedil;&ecirc;&atilde;
&ntilde; = ~N + &iacute;&Auml;N
I

&oacute; = ~ O + &iacute;&Auml;O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
(&ntilde; I &oacute; ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; ~&aring;&oacute; &igrave;&aring;&acirc;&aring;&ccedil;&iuml;&aring; &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I
(~N I ~ O ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; ~ &acirc;&aring;&ccedil;&iuml;&aring; &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I
(&Auml;N I &Auml;O ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I
&iacute; &aacute;&euml; ~ &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 107.
635. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; c&ecirc;&ccedil;&atilde; ~ m&ccedil;&aacute;&aring;&iacute; q&ccedil; ~ i&aacute;&aring;&Eacute;
q&Uuml;&Eacute; &Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Ntilde;&ecirc;&ccedil;&atilde; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; m(~ I &Auml;) &iacute;&ccedil; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;
^&ntilde; + _&oacute; + ` = M &aacute;&euml;
^~ + _&Auml; + `
&Ccedil;=
K
^O + _O
Figure 108.
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CHAPTER 7. ANALYTIC GEOMETRY
636. m~&ecirc;~&auml;&auml;&Eacute;&auml; i&aacute;&aring;&Eacute;&euml;
q&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml; &oacute; = &acirc; N&ntilde; + &Auml;N ~&aring;&Ccedil; &oacute; = &acirc; O &ntilde; + &Auml;O ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &aacute;&Ntilde;
&acirc;N = &acirc; O K
q&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml; ^N&ntilde; + _N&oacute; + `N = M ~&aring;&Ccedil; ^ O &ntilde; + _O &oacute; + ` O = M ~&ecirc;&Eacute;
&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &aacute;&Ntilde;
^N _N
=
K
^ O _O
Figure 109.
637. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; i&aacute;&aring;&Eacute;&euml;
q&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml; &oacute; = &acirc; N&ntilde; + &Auml;N ~&aring;&Ccedil; &oacute; = &acirc; O &ntilde; + &Auml;O ~&ecirc;&Eacute; &eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; &aacute;&Ntilde;
N
&acirc;O = −
&ccedil;&ecirc;I &Eacute;&egrave;&igrave;&aacute;&icirc;~&auml;&Eacute;&aring;&iacute;&auml;&oacute;I &acirc; N&acirc; O = −N K
&acirc;N
q&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml; ^N&ntilde; + _N&oacute; + `N = M ~&aring;&Ccedil; ^ O &ntilde; + _ O &oacute; + ` O = M ~&ecirc;&Eacute;
&eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; &aacute;&Ntilde;
^N^ O + _N_ O = M K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 110.
638. ^&aring;&Ouml;&auml;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; i&aacute;&aring;&Eacute;&euml;
&acirc; − &acirc;N
&iacute;~&aring; ϕ = O
I
N + &acirc; N&acirc; O
^N^ O + _N_ O
K
&Aring;&ccedil;&euml; ϕ =
^NO + _NO ⋅ ^ OO + _ OO
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 111.
639. f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; q&iuml;&ccedil; i&aacute;&aring;&Eacute;&euml;
f&Ntilde; &iacute;&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml; ^N&ntilde; + _N&oacute; + `N = M ~&aring;&Ccedil; ^ O &ntilde; + _ O &oacute; + ` O = M &aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;I &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &eacute;&ccedil;&aacute;&aring;&iacute; &Uuml;~&euml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
− ^N` O + ^ O`N
− `N_ O + ` O_N
K
&ntilde;M =
I &oacute;M =
^N_ O − ^ O_N
^N_ O − ^ O_N
7.4 Circle
o~&Ccedil;&aacute;&igrave;&euml;W o
`&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W (~ I &Auml;)
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ntilde;I &oacute;I &ntilde; N I &oacute; N I &pound;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I _I `I aI bI cI &iacute;
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640. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ `&aacute;&ecirc;&Aring;&auml;&Eacute; `&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil; ~&iacute; &iacute;&Uuml;&Eacute; l&ecirc;&aacute;&Ouml;&aacute;&aring; Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;
c&ccedil;&ecirc;&atilde;F
&ntilde; O + &oacute; O = oO
Figure 112.
641. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ `&aacute;&ecirc;&Aring;&auml;&Eacute; `&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil; ~&iacute; ^&aring;&oacute; m&ccedil;&aacute;&aring;&iacute; (~I &Auml;)
(&ntilde; − ~ )O + (&oacute; − &Auml;)O = o O
Figure 113.
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CHAPTER 7. ANALYTIC GEOMETRY
642. q&Uuml;&ecirc;&Eacute;&Eacute; m&ccedil;&aacute;&aring;&iacute; c&ccedil;&ecirc;&atilde;
&ntilde;O + &oacute;O &ntilde; &oacute; N
&ntilde; NO + &oacute; NO
&ntilde;N
&oacute;N N
&ntilde; +&oacute;
&ntilde; +&oacute;
&ntilde;O
&ntilde;P
&oacute;O N
&oacute;P N
O
O
O
P
O
O
O
P
=M
Figure 114.
643. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&ccedil;&ecirc;&atilde;
&ntilde; = o &Aring;&ccedil;&euml; &iacute;
I M ≤ &iacute; ≤ Oπ K

&oacute; = o &euml;&aacute;&aring; &iacute;
644. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde;
^&ntilde; O + ^&oacute; O + a&ntilde; + b&oacute; + c = M E^ &aring;&ccedil;&aring;&ograve;&Eacute;&ecirc;&ccedil;I aO + b O &gt; Q ^c FK
q&Uuml;&Eacute; &Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute; &Uuml;~&euml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; (~ I &Auml;) I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
a
b
~=−
I &Auml;=−
K
O^
O^
q&Uuml;&Eacute; &ecirc;~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&aacute;&ecirc;&Aring;&auml;&Eacute; &aacute;&euml;
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CHAPTER 7. ANALYTIC GEOMETRY
o=
aO + b O − Q ^c
K
O^
7.5 Ellipse
p&Eacute;&atilde;&aacute;&atilde;~&agrave;&ccedil;&ecirc; ~&ntilde;&aacute;&euml;W ~
p&Eacute;&atilde;&aacute;&atilde;&aacute;&aring;&ccedil;&ecirc; ~&ntilde;&aacute;&euml;W &Auml;
c&ccedil;&Aring;&aacute;W cN (− &Aring;I M) I cO (&Aring;I M)
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&Aring;&aacute;W O&Aring;
b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;W &Eacute;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I _I `I aI bI cI &iacute;
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W i
^&ecirc;&Eacute;~W p
645. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~&aring; b&auml;&auml;&aacute;&eacute;&euml;&Eacute; Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil; c&ccedil;&ecirc;&atilde;F
&ntilde;O &oacute;O
+ =N
~ O &Auml;O
Figure 115.
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CHAPTER 7. ANALYTIC GEOMETRY
646. &ecirc;N + &ecirc;O = O~ I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &ecirc;N I &ecirc;O ~&ecirc;&Eacute; &Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;&euml; &Ntilde;&ecirc;&ccedil;&atilde; ~&aring;&oacute; &eacute;&ccedil;&aacute;&aring;&iacute; m(&ntilde; I &oacute; ) &ccedil;&aring;
&iacute;&Uuml;&Eacute; &Eacute;&auml;&auml;&aacute;&eacute;&euml;&Eacute; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &iacute;&iuml;&ccedil; &Ntilde;&ccedil;&Aring;&aacute;K
Figure 116.
647. ~ O = &Auml;O + &Aring; O
648. b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;
&Aring;
&Eacute; = &lt;N
~
649. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; a&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;
~
~O
&ntilde;=&plusmn; =&plusmn;
&Eacute;
&Aring;
650. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&ccedil;&ecirc;&atilde;
&ntilde; = ~ &Aring;&ccedil;&euml; &iacute;
I M ≤ &iacute; ≤ Oπ K

&oacute; = &Auml; &euml;&aacute;&aring; &iacute;
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651. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde;
^&ntilde; O + _&ntilde;&oacute; + `&oacute; O + a&ntilde; + b&oacute; + c = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; _ O − Q ^` &lt; M K
652. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde; &iuml;&aacute;&iacute;&Uuml; ^&ntilde;&Eacute;&euml; m~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; ^&ntilde;&Eacute;&euml;
^&ntilde; O + `&oacute; O + a&ntilde; + b&oacute; + c = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ^` &gt; M K
653. `&aacute;&ecirc;&Aring;&igrave;&atilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;
i = Q~b(&Eacute; ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; b &aacute;&euml; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&iacute;&Eacute; &Eacute;&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde;
&iacute;&Uuml;&Eacute; &euml;&Eacute;&Aring;&ccedil;&aring;&Ccedil; &acirc;&aacute;&aring;&Ccedil;K
654. ^&eacute;&eacute;&ecirc;&ccedil;&ntilde;&aacute;&atilde;~&iacute;&Eacute; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; `&aacute;&ecirc;&Aring;&igrave;&atilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;
i = π NKR(~ + &Auml;) − ~&Auml; I
(
i = π O(~ O + &Auml;O ) K
)
655. p = π~&Auml;
7.6 Hyperbola
q&ecirc;~&aring;&euml;&icirc;&Eacute;&ecirc;&euml;&Eacute; ~&ntilde;&aacute;&euml;W ~
`&ccedil;&aring;&agrave;&igrave;&Ouml;~&iacute;&Eacute; ~&ntilde;&aacute;&euml;W &Auml;
c&ccedil;&Aring;&aacute;W cN (− &Aring;I M) I cO (&Aring;I M)
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&Aring;&aacute;W O&Aring;
b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;W &Eacute;
^&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&Eacute;&euml;W &euml;I &iacute;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I _I `I aI bI cI &iacute;I &acirc;
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CHAPTER 7. ANALYTIC GEOMETRY
656. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~ Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil; c&ccedil;&ecirc;&atilde;F
&ntilde;O &oacute;O
− =N
~ O &Auml;O
Figure 117.
657.
&ecirc;N − &ecirc;O = O~ I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &ecirc;N I &ecirc;O ~&ecirc;&Eacute; &Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;&euml; &Ntilde;&ecirc;&ccedil;&atilde; ~&aring;&oacute; &eacute;&ccedil;&aacute;&aring;&iacute; m(&ntilde; I &oacute; ) &ccedil;&aring;
&iacute;&Uuml;&Eacute; &Uuml;&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~ &iacute;&ccedil; &iacute;&Uuml;&Eacute; &iacute;&iuml;&ccedil; &Ntilde;&ccedil;&Aring;&aacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 118.
658. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; ^&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&Eacute;&euml;
&Auml;
&oacute;=&plusmn; &ntilde;
~
659. &Aring; O = ~ O + &Auml;O
660. b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;
&Aring;
&Eacute; = &gt;N
~
661. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; a&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;
~
~O
&ntilde;=&plusmn; =&plusmn;
&Eacute;
&Aring;
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CHAPTER 7. ANALYTIC GEOMETRY
662. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; o&aacute;&Ouml;&Uuml;&iacute; _&ecirc;~&aring;&Aring;&Uuml; &ccedil;&Ntilde; ~ e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~
&ntilde; = ~ &Aring;&ccedil;&euml;&Uuml; &iacute;
I M ≤ &iacute; ≤ Oπ K

&oacute; = &Auml; &euml;&aacute;&aring;&Uuml; &iacute;
663. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde;
^&ntilde; O + _&ntilde;&oacute; + `&oacute; O + a&ntilde; + b&oacute; + c = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; _ O − Q ^` &gt; M K
664. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde; &iuml;&aacute;&iacute;&Uuml; ^&ntilde;&Eacute;&euml; m~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; ^&ntilde;&Eacute;&euml;
^&ntilde; O + `&oacute; O + a&ntilde; + b&oacute; + c = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ^` &lt; M K
665. ^&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&aacute;&Aring; c&ccedil;&ecirc;&atilde;
&Eacute;O
&ntilde;&oacute; = I
Q
&ccedil;&ecirc;
&Eacute;O
&acirc;
&oacute; = I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; &acirc; = K
&ntilde;
Q
f&aring; &iacute;&Uuml;&aacute;&euml; &Aring;~&euml;&Eacute; I &iacute;&Uuml;&Eacute; ~&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&Eacute;&euml; &Uuml;~&icirc;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ntilde; = M ~&aring;&Ccedil;
&oacute; =MK
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Figure 119.
7.7 Parabola
c&ccedil;&Aring;~&auml; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;W &eacute;
c&ccedil;&Aring;&igrave;&euml;W c
s&Eacute;&ecirc;&iacute;&Eacute;&ntilde;W j(&ntilde; M I &oacute; M )
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I _I `I aI bI cI &eacute;I ~I &Auml;I &Aring;
666. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ m~&ecirc;~&Auml;&ccedil;&auml;~ Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil; c&ccedil;&ecirc;&atilde;F
&oacute; O = O&eacute;&ntilde;
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 120.
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&ntilde;
&eacute;
&ntilde;=− I
O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&Aring;&igrave;&euml;
&eacute; 
c I M  I
O 
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&ecirc;&iacute;&Eacute;&ntilde;
j(MI M) K
667. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde;
^&ntilde; O + _&ntilde;&oacute; + `&oacute; O + a&ntilde; + b&oacute; + c = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; _ O − Q ^` = M K
N
K
O~
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&ntilde;
668. &oacute; = ~&ntilde; O I &eacute; =
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CHAPTER 7. ANALYTIC GEOMETRY
&eacute;
&oacute;=− I
O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&Aring;&igrave;&euml;
 &eacute;
c MI  I
 O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&ecirc;&iacute;&Eacute;&ntilde;
j(MI M) K
Figure 121.
669. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde;I ^&ntilde;&aacute;&euml; m~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &oacute;-~&ntilde;&aacute;&euml;
^&ntilde; O + a&ntilde; + b&oacute; + c = M E^I b &aring;&ccedil;&aring;&ograve;&Eacute;&ecirc;&ccedil;FI
N
&oacute; = ~&ntilde; O + &Auml;&ntilde; + &Aring; I &eacute; = K
O~
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&ntilde;
&eacute;
&oacute; = &oacute;M − I
O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&Aring;&igrave;&euml;
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CHAPTER 7. ANALYTIC GEOMETRY
&eacute;

c &ntilde; M I &oacute; M +  I
O

`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&ecirc;&iacute;&Eacute;&ntilde;
&Auml;
Q~&Aring; − &Auml;O
&ntilde; M = − I &oacute; M = ~&ntilde; OM + &Auml;&ntilde; M + &Aring; =
K
O~
Q~
Figure 122.
7.8 Three-Dimensional Coordinate System
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ntilde; M I &oacute; M I &ograve; M I &ntilde; N I &oacute; N I &ograve; N I &pound;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W λ
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&iuml;&ccedil; &eacute;&ccedil;&aacute;&aring;&iacute;&euml;W &Ccedil;
^&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute;W s
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CHAPTER 7. ANALYTIC GEOMETRY
670. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; m&ccedil;&aacute;&aring;&iacute;&euml;
&Ccedil; = ^_ =
(&ntilde; O − &ntilde; N )O + (&oacute; O − &oacute;N )O + (&ograve; O − &ograve;N )O
Figure 123.
671. a&aacute;&icirc;&aacute;&Ccedil;&aacute;&aring;&Ouml; ~ i&aacute;&aring;&Eacute; p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute; &aacute;&aring; &iacute;&Uuml;&Eacute; o~&iacute;&aacute;&ccedil; λ
&ntilde; + λ&ntilde; O
&oacute; + λ&oacute; O
&ograve; + λ&ograve; O
I &oacute;M = N
I &ograve;M = N
I
&ntilde;M = N
N+ λ
N+ λ
N+ λ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
^`
λ=
I λ ≠ −N K
`_
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Figure 124.
Figure 125.
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CHAPTER 7. ANALYTIC GEOMETRY
672. j&aacute;&Ccedil;&eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; ~ i&aacute;&aring;&Eacute; p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;
&ntilde; + &ntilde;O
&oacute; + &oacute;O
&ograve; +&ograve;
I &ograve;M = N O I λ = N K
&ntilde;M = N
I &oacute;M = N
O
O
O
673. ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;
q&Uuml;&Eacute; ~&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ &iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute; &iuml;&aacute;&iacute;&Uuml; &icirc;&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml; mN (&ntilde; N I &oacute; N I &ograve; N ) I
mO (&ntilde; O I &oacute; O I &ograve; O ) I ~&aring;&Ccedil; mP (&ntilde; P I &oacute; P I &ograve; P ) &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
p=
N
O
O
&ntilde;N N
&oacute;O
&ograve;O N + &ograve;O
&ntilde;O N + &ntilde;O
&oacute;O N K
&oacute;P
&ograve;P
&ntilde;P
&oacute;P
&ograve;P
N
&ntilde;N
&ntilde;P
&oacute;N N
O
&ograve;N N
N
&ograve;N
O
&oacute;N
N
674. s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ q&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;
q&Uuml;&Eacute; &icirc;&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ &iacute;&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring; &iuml;&aacute;&iacute;&Uuml; &icirc;&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml; mN (&ntilde; N I &oacute; N I &ograve; N ) I
mO (&ntilde; O I &oacute; O I &ograve; O ) I mP (&ntilde; P I &oacute; P I &ograve; P ) I ~&aring;&Ccedil; mQ (&ntilde; Q I &oacute; Q I &ograve; Q ) &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
&ntilde; N &oacute;N &ograve;N N
&oacute;O
&oacute;P
&oacute;Q
&ograve;O N
I
&ograve;P N
&ograve;Q N
&ntilde;N − &ntilde; Q
N
s = &plusmn; &ntilde;O − &ntilde;Q
S
&ntilde;P − &ntilde; Q
&oacute;N − &oacute; Q
&ograve;N − &ograve; Q
&oacute;O − &oacute;Q
&oacute;P − &oacute; Q
&ograve;O − &ograve;Q K
&ograve;P − &ograve;Q
s=&plusmn;
N &ntilde;O
S &ntilde;P
&ntilde;Q
&ccedil;&ecirc;
k&ccedil;&iacute;&Eacute;W t&Eacute; &Aring;&Uuml;&ccedil;&ccedil;&euml;&Eacute; &iacute;&Uuml;&Eacute; &euml;&aacute;&Ouml;&aring; EHF &ccedil;&ecirc; E&yen;F &euml;&ccedil; &iacute;&Uuml;~&iacute; &iacute;&ccedil; &Ouml;&Eacute;&iacute; ~ &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;
~&aring;&euml;&iuml;&Eacute;&ecirc; &Ntilde;&ccedil;&ecirc; &icirc;&ccedil;&auml;&igrave;&atilde;&Eacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 126.
7.9 Plane
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ntilde;I &oacute;I &ograve;I &ntilde; M I &oacute; M I &ograve; M I &ntilde; N I &oacute; N I &ograve; N I &pound;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I _I `I aI ^N I ^ O I ~I &Auml;I &Aring;I ~N I ~ O I λ I &eacute;I &iacute;I &pound;
r r r
k&ccedil;&ecirc;&atilde;~&auml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W &aring; I &aring;N I &aring; O
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;W &Aring;&ccedil;&euml; α I &Aring;&ccedil;&euml; β I &Aring;&ccedil;&euml; γ
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Ntilde;&ecirc;&ccedil;&atilde; &eacute;&ccedil;&aacute;&aring;&iacute; &iacute;&ccedil; &eacute;&auml;~&aring;&Eacute;W &Ccedil;
675. d&Eacute;&aring;&Eacute;&ecirc;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ m&auml;~&aring;&Eacute;
^&ntilde; + _&oacute; + `&ograve; + a = M
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CHAPTER 7. ANALYTIC GEOMETRY
676. k&ccedil;&ecirc;&atilde;~&auml; s&Eacute;&Aring;&iacute;&ccedil;&ecirc; &iacute;&ccedil; ~ m&auml;~&aring;&Eacute;
r
q&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &aring; (^I _I ` ) &aacute;&euml; &aring;&ccedil;&ecirc;&atilde;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;
^&ntilde; + _&oacute; + `&ograve; + a = M K
Figure 127.
677. m~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; `~&euml;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ m&auml;~&aring;&Eacute;
^&ntilde; + _&oacute; + `&ograve; + a = M
f&Ntilde;
f&Ntilde;
f&Ntilde;
f&Ntilde;
^ = M I &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; &aacute;&euml; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml;K
_ = M I &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; &aacute;&euml; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &oacute;-~&ntilde;&aacute;&euml;K
` = M I &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; &aacute;&euml; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &ograve;-~&ntilde;&aacute;&euml;K
a = M I &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; &auml;&aacute;&Eacute;&euml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;K
f&Ntilde; ^ = _ = M I &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; &aacute;&euml; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K
f&Ntilde; _ = ` = M I &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; &aacute;&euml; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &oacute;&ograve;-&eacute;&auml;~&aring;&Eacute;K
f&Ntilde; ^ = ` = M I &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; &aacute;&euml; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &ntilde;&ograve;-&eacute;&auml;~&aring;&Eacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
678. m&ccedil;&aacute;&aring;&iacute; a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; c&ccedil;&ecirc;&atilde;
^(&ntilde; − &ntilde; M ) + _(&oacute; − &oacute; M ) + `(&ograve; − &ograve; M ) = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; m(&ntilde; M I &oacute; M I &ograve; M ) &auml;&aacute;&Eacute;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;I ~&aring;&Ccedil; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; (^I _I ` ) &aacute;&euml; &aring;&ccedil;&ecirc;&atilde;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;K
Figure 128.
679. f&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute; c&ccedil;&ecirc;&atilde;
&ntilde; &oacute; &ograve;
+ + =N
~ &Auml; &Aring;
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 129.
680. q&Uuml;&ecirc;&Eacute;&Eacute; m&ccedil;&aacute;&aring;&iacute; c&ccedil;&ecirc;&atilde;
&ntilde; − &ntilde;P &oacute; − &oacute;P &ograve; − &ograve;P
&ntilde; N − &ntilde; P &oacute;N − &oacute; P &ograve;N − &ograve; P = M I
&ntilde; O − &ntilde; P &oacute; O − &oacute;P &ograve; O − &ograve;P
&ccedil;&ecirc;
&ntilde; &oacute; &ograve; N
&ntilde;N &oacute;N &ograve;N N
=MK
&ntilde;O &oacute;O &ograve;O N
&ntilde;P &oacute;P &ograve;P N
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 130.
681. k&ccedil;&ecirc;&atilde;~&auml; c&ccedil;&ecirc;&atilde;
&ntilde; &Aring;&ccedil;&euml; α + &oacute; &Aring;&ccedil;&euml; β + &ograve; &Aring;&ccedil;&euml; γ − &eacute; = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &eacute; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; &Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Ntilde;&ecirc;&ccedil;&atilde; &iacute;&Uuml;&Eacute; &ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring; &iacute;&ccedil;
&iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute; I ~&aring;&Ccedil; &Aring;&ccedil;&euml; α I &Aring;&ccedil;&euml; β I &Aring;&ccedil;&euml; γ ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;
&ccedil;&Ntilde; ~&aring;&oacute; &auml;&aacute;&aring;&Eacute; &aring;&ccedil;&ecirc;&atilde;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 131.
682. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&ccedil;&ecirc;&atilde;
&ntilde; = &ntilde; N + ~N&euml; + ~ O &iacute;

&oacute; = &oacute; N + &Auml;N&euml; + &Auml;O &iacute; I
&ograve; = &ograve; + &Aring; &euml; + &Aring; &iacute;
N
N
O

&iuml;&Uuml;&Eacute;&ecirc;&Eacute; (&ntilde; I &oacute; I &ograve; ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; ~&aring;&oacute; &igrave;&aring;&acirc;&aring;&ccedil;&iuml;&aring; &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&aring;
&iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; I &iacute;&Uuml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; m(&ntilde; N I &oacute; N I &ograve; N ) &auml;&aacute;&Eacute;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;I &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;
(~N I &Auml;N I &Aring;N ) ~&aring;&Ccedil; (~ O I &Auml;O I &Aring; O ) ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 132.
683. a&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml; ^&aring;&Ouml;&auml;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; m&auml;~&aring;&Eacute;&euml;
f&Ntilde; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;&euml; ~&ecirc;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
^N&ntilde; + _N&oacute; + `N&ograve; + aN = M I
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M I
&iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml; ~&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute;&atilde; &aacute;&euml;
r r
&aring;N ⋅ &aring; O
^N^ O + _N_ O + `N` O
&Aring;&ccedil;&euml; ϕ = r r =
K
&aring;N ⋅ &aring; O
^NO + _NO + `NO ⋅ ^ OO + _ OO + ` OO
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 133.
684. m~&ecirc;~&auml;&auml;&Eacute;&auml; m&auml;~&aring;&Eacute;&euml;
q&iuml;&ccedil; &eacute;&auml;~&aring;&Eacute;&euml; ^N&ntilde; + _N&oacute; + `N&ograve; + aN = M ~&aring;&Ccedil;
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &aacute;&Ntilde;
^N _N `N
=
=
K
^O _O `O
685. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; m&auml;~&aring;&Eacute;&euml;
q&iuml;&ccedil; &eacute;&auml;~&aring;&Eacute;&euml; ^N&ntilde; + _N&oacute; + `N&ograve; + aN = M ~&aring;&Ccedil;
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M ~&ecirc;&Eacute; &eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; &aacute;&Ntilde;
^N^ O + _N_ O + `N` O = M K
686. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ m&auml;~&aring;&Eacute; q&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml; m(&ntilde; N I &oacute; N I &ograve; N ) ~&aring;&Ccedil; m~&ecirc;~&auml;&auml;&Eacute;&auml; q&ccedil;
&iacute;&Uuml;&Eacute; s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; (~N I &Auml;N I &Aring;N ) ~&aring;&Ccedil; (~ O I &Auml;O I &Aring; O ) Ec&aacute;&Ouml;KNPOF
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CHAPTER 7. ANALYTIC GEOMETRY
&ntilde; − &ntilde;N
~N
~O
&oacute; − &oacute;N &ograve; − &ograve;N
&Auml;N
&Aring;N = M
&Auml;O
&Aring;O
687. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ m&auml;~&aring;&Eacute; q&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml; mN (&ntilde; N I &oacute; N I &ograve; N ) ~&aring;&Ccedil; mO (&ntilde; O I &oacute; O I &ograve; O ) I
~&aring;&Ccedil; m~&ecirc;~&auml;&auml;&Eacute;&auml; q&ccedil; &iacute;&Uuml;&Eacute; s&Eacute;&Aring;&iacute;&ccedil;&ecirc; (~ I &Auml;I &Aring; )
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
&ntilde; O − &ntilde;N
~
&oacute; O − &oacute;N &ograve; O − &ograve;N = M
&Auml;
&Aring;
Figure 134.
688. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; c&ecirc;&ccedil;&atilde; ~ m&ccedil;&aacute;&aring;&iacute; q&ccedil; ~ m&auml;~&aring;&Eacute;
q&Uuml;&Eacute; &Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute; &Ntilde;&ecirc;&ccedil;&atilde; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; mN (&ntilde; N I &oacute; N I &ograve; N ) &iacute;&ccedil; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;
^&ntilde; + _&oacute; + `&ograve; + a = M &aacute;&euml;
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CHAPTER 7. ANALYTIC GEOMETRY
&Ccedil;=
^&ntilde; N + _&oacute; N + `&ograve;N + a
^O + _O + `O
K
Figure 135.
689. f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; q&iuml;&ccedil; m&auml;~&aring;&Eacute;&euml;
f&Ntilde; &iacute;&iuml;&ccedil; &eacute;&auml;~&aring;&Eacute;&euml; ^N&ntilde; + _N&oacute; + `N&ograve; + aN = M ~&aring;&Ccedil;
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M &aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;I &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;
&auml;&aacute;&aring;&Eacute; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
&ntilde; = &ntilde; N + ~&iacute;

&oacute; = &oacute; N + &Auml;&iacute; I
&ograve; = &ograve; + &Aring;&iacute;
N

&ccedil;&ecirc;
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
=
=
I
&Auml;
&Aring;
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
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CHAPTER 7. ANALYTIC GEOMETRY
~=
_N
_O
&Auml;
aN
aO
&Aring;
aN
aO
~
aN
aO
&ntilde;N =
&oacute;N =
&ograve;N =
`N
`
I &Auml;= N
`O
`O
^N
^
I &Aring;= N
^O
^O
_N
I
_O
`N
a _N
−&Aring; N
`O
aO _ O
I
O
O
~ + &Auml; + &Aring;O
^N
a `N
−~ N
^O
aO ` O
I
O
O
~ + &Auml; + &Aring;O
_N
a ^N
−&Auml; N
_O
aO ^ O
K
O
O
~ + &Auml; + &Aring;O
7.10 Straight Line in Space
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ntilde;I &oacute;I &ograve;I &ntilde; N I &oacute; N I &ograve; N I &pound;
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;W &Aring;&ccedil;&euml; α I &Aring;&ccedil;&euml; β I &Aring;&ccedil;&euml; γ
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I _I `I aI ~I &Auml;I &Aring;I ~N I ~ O I &iacute;I &pound;
r r r
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; &ccedil;&Ntilde; ~ &auml;&aacute;&aring;&Eacute;W &euml; I &euml;N I &euml;O
r
k&ccedil;&ecirc;&atilde;~&auml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &iacute;&ccedil; ~ &eacute;&auml;~&aring;&Eacute;W &aring;
^&aring;&Ouml;&auml;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml;W ϕ
690. m&ccedil;&aacute;&aring;&iacute; a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; c&ccedil;&ecirc;&atilde; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ i&aacute;&aring;&Eacute;
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
=
=
I
&Auml;
&Aring;
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; mN (&ntilde; N I &oacute; N I &ograve; N ) &auml;&aacute;&Eacute;&euml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I ~&aring;&Ccedil; (~ I &Auml;I &Aring; ) &aacute;&euml;
&iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;K
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 136.
691. q&iuml;&ccedil; m&ccedil;&aacute;&aring;&iacute; c&ccedil;&ecirc;&atilde;
&ntilde; − &ntilde;N
&oacute; − &oacute;N
&ograve; − &ograve;N
=
=
&ntilde; O − &ntilde; N &oacute; O − &oacute;N &ograve; O − &ograve;N
Figure 137.
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CHAPTER 7. ANALYTIC GEOMETRY
692. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; c&ccedil;&ecirc;&atilde;
&ntilde; = &ntilde; N + &iacute; &Aring;&ccedil;&euml; α

&oacute; = &oacute; N + &iacute; &Aring;&ccedil;&euml; β I
&ograve; = &ograve; + &iacute; &Aring;&ccedil;&euml; γ
N

&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; mN (&ntilde; N I &oacute; N I &ograve; N ) &auml;&aacute;&Eacute;&euml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; &auml;&aacute;&aring;&Eacute;I
&Aring;&ccedil;&euml; α I &Aring;&ccedil;&euml; β I &Aring;&ccedil;&euml; γ ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;
&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I &iacute;&Uuml;&Eacute; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc; &iacute; &aacute;&euml; ~&aring;&oacute; &ecirc;&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;K
Figure 138.
693. ^&aring;&Ouml;&auml;&Eacute; _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; i&aacute;&aring;&Eacute;&euml;
r r
&euml; ⋅&euml;
~N~ O + &Auml;N&Auml;O + &Aring;N&Aring; O
&Aring;&ccedil;&euml; ϕ = r N rO =
O
&euml;N ⋅ &euml;O
~N + &Auml;NO + &Aring;NO ⋅ ~ OO + &Auml;OO + &Aring; OO
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 139.
694. m~&ecirc;~&auml;&auml;&Eacute;&auml; i&aacute;&aring;&Eacute;&euml;
q&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml; ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &aacute;&Ntilde;
r r
&euml;N &ouml;&ouml; &euml;O I
&ccedil;&ecirc;
~N &Auml;N &Aring;N
= = K
~ O &Auml;O &Aring; O
695. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; i&aacute;&aring;&Eacute;&euml;
q&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml; ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &aacute;&Ntilde;
r r
&euml;N ⋅ &euml;O = M I
&ccedil;&ecirc;
~N~ O + &Auml;N&Auml;O + &Aring;N&Aring; O = M K
696. f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; q&iuml;&ccedil; i&aacute;&aring;&Eacute;&euml;
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
q&iuml;&ccedil; &auml;&aacute;&aring;&Eacute;&euml;
=
=
~&aring;&Ccedil;
~N
&Auml;N
&Aring;N
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CHAPTER 7. ANALYTIC GEOMETRY
&ntilde; − &ntilde;O &oacute; − &oacute;O &ograve; − &ograve;O
=
=
&aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute; &aacute;&Ntilde;
~O
&Auml;O
&Aring;O
&ntilde; O − &ntilde; N &oacute; O − &oacute;N &ograve; O − &ograve;N
~N
&Auml;N
&Aring;N = M K
~O
&Auml;O
&Aring;O
697. m~&ecirc;~&auml;&auml;&Eacute;&auml; i&aacute;&aring;&Eacute; ~&aring;&Ccedil; m&auml;~&aring;&Eacute;
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
q&Uuml;&Eacute; &euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; &auml;&aacute;&aring;&Eacute;
=
=
~&aring;&Ccedil; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;
&Auml;
&Aring;
~
^&ntilde; + _&oacute; + `&ograve; + a = M ~&ecirc;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml; &aacute;&Ntilde;
r r
&aring;⋅ &euml; = M I
&ccedil;&ecirc;
^~ + _&Auml; + `&Aring; = M K
Figure 140.
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CHAPTER 7. ANALYTIC GEOMETRY
698. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; i&aacute;&aring;&Eacute; ~&aring;&Ccedil; m&auml;~&aring;&Eacute;
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
q&Uuml;&Eacute; &euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute; &auml;&aacute;&aring;&Eacute;
=
=
~&aring;&Ccedil; &iacute;&Uuml;&Eacute; &eacute;&auml;~&aring;&Eacute;
&Auml;
&Aring;
~
^&ntilde; + _&oacute; + `&ograve; + a = M ~&ecirc;&Eacute; &eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc; &aacute;&Ntilde;
r r
&aring; &ouml;&ouml; &euml; I
&ccedil;&ecirc;
^ _ `
= = K
~ &Auml; &Aring;
Figure 141.
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &egrave;&igrave;~&Ccedil;&ecirc;&aacute;&Aring; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;&euml;W &ntilde;I &oacute;I &ograve;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I _I `I ~I &Auml;I &Aring;I &acirc; N I &acirc; O I &acirc; P I &pound;
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CHAPTER 7. ANALYTIC GEOMETRY
699. d&Eacute;&aring;&Eacute;&ecirc;~&auml; n&igrave;~&Ccedil;&ecirc;~&iacute;&aacute;&Aring; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
^&ntilde; O + _&oacute; O + `&ograve; O + Oc&oacute;&ograve; + Od&ograve;&ntilde; + Oe&ntilde;&oacute; + Om&ntilde; + On&oacute; + Oo&ograve; + a = M
700. `&auml;~&euml;&euml;&aacute;&Ntilde;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; n&igrave;~&Ccedil;&ecirc;&aacute;&Aring; p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;&euml;
`~&euml;&Eacute;
N
O
P
Q
R
S
T
U
V
NM
NN
NO
NP
NQ
NR
NS
NT
o~&aring;&acirc;E&Eacute;F
P
P
P
P
P
P
O
O
O
O
O
O
O
N
N
N
N
o~&aring;&acirc;EbF
Q
Q
Q
Q
P
P
Q
Q
P
P
P
O
O
P
O
O
N
∆
&lt;M
&gt;M
&gt;M
&lt;M
&lt;M
&gt;M
&acirc; &euml;&aacute;&Ouml;&aring;&euml;
p~&atilde;&Eacute;
p~&atilde;&Eacute;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;
p~&atilde;&Eacute;
p~&atilde;&Eacute;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;
p~&atilde;&Eacute;
p~&atilde;&Eacute;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;
p~&atilde;&Eacute;
q&oacute;&eacute;&Eacute; &ccedil;&Ntilde; p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;
o&Eacute;~&auml; b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil;
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil;
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil; &ccedil;&Ntilde; N p&Uuml;&Eacute;&Eacute;&iacute;
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil; &ccedil;&Ntilde; O p&Uuml;&Eacute;&Eacute;&iacute;&euml;
o&Eacute;~&auml; n&igrave;~&Ccedil;&ecirc;&aacute;&Aring; `&ccedil;&aring;&Eacute;
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; n&igrave;~&Ccedil;&ecirc;&aacute;&Aring; `&ccedil;&aring;&Eacute;
b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring; m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;
o&Eacute;~&auml; b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;
o&Eacute;~&auml; f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml; m&auml;~&aring;&Eacute;&euml;
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml; m&auml;~&aring;&Eacute;&euml;
m~&ecirc;~&Auml;&ccedil;&auml;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;
o&Eacute;~&auml; m~&ecirc;~&auml;&auml;&Eacute;&auml; m&auml;~&aring;&Eacute;&euml;
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; m~&ecirc;~&auml;&auml;&Eacute;&auml; m&auml;~&aring;&Eacute;&euml;
`&ccedil;&aacute;&aring;&Aring;&aacute;&Ccedil;&Eacute;&aring;&iacute; m&auml;~&aring;&Eacute;&euml;
e&Eacute;&ecirc;&Eacute;
^ e n m


 ^ e d


 e _ c n
I ∆ = &Ccedil;&Eacute;&iacute;(b ) I
&Eacute; = e _ c I b = 
d c ` o
 d c `




 m n o a


&acirc; N I &acirc; O I &acirc; P ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &ecirc;&ccedil;&ccedil;&iacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;I
^−&ntilde;
e
d
e
_−&ntilde;
c =MK
d
c
`−&ntilde;
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CHAPTER 7. ANALYTIC GEOMETRY
701. o&Eacute;~&auml; b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil; E`~&euml;&Eacute; NF
&ntilde;O &oacute;O &ograve;O
+ + =N
~ O &Auml;O &Aring; O
Figure 142.
702. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil; E`~&euml;&Eacute; OF
&ntilde;O &oacute;O &ograve;O
+ + = −N
~ O &Auml;O &Aring; O
703. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil; &ccedil;&Ntilde; N p&Uuml;&Eacute;&Eacute;&iacute; E`~&euml;&Eacute; PF
&ntilde;O &oacute;O &ograve;O
+ − =N
~ O &Auml;O &Aring; O
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 143.
704. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil; &ccedil;&Ntilde; O p&Uuml;&Eacute;&Eacute;&iacute;&euml; E`~&euml;&Eacute; QF
&ntilde;O &oacute;O &ograve;O
+ − = −N
~ O &Auml;O &Aring; O
Figure 144.
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CHAPTER 7. ANALYTIC GEOMETRY
705. o&Eacute;~&auml; n&igrave;~&Ccedil;&ecirc;&aacute;&Aring; `&ccedil;&aring;&Eacute; E`~&euml;&Eacute; RF
&ntilde;O &oacute;O &ograve;O
+ − =M
~ O &Auml;O &Aring; O
Figure 145.
706. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; n&igrave;~&Ccedil;&ecirc;&aacute;&Aring; `&ccedil;&aring;&Eacute; E`~&euml;&Eacute; SF
&ntilde;O &oacute;O &ograve;O
+ + =M
~ O &Auml;O &Aring; O
707. b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring; m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil; E`~&euml;&Eacute; TF
&ntilde;O &oacute;O
+ −&ograve; = M
~ O &Auml;O
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 146.
708. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil; E`~&euml;&Eacute; UF
&ntilde;O &oacute;O
− −&ograve; = M
~ O &Auml;O
Figure 147.
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CHAPTER 7. ANALYTIC GEOMETRY
709. o&Eacute;~&auml; b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc; E`~&euml;&Eacute; VF
&ntilde;O &oacute;O
+ =N
~ O &Auml;O
Figure 148.
710. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc; E`~&euml;&Eacute; NMF
&ntilde;O &oacute;O
+ = −N
~ O &Auml;O
711. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc; E`~&euml;&Eacute; NNF
&ntilde;O &oacute;O
− =N
~ O &Auml;O
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 149.
712. o&Eacute;~&auml; f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml; m&auml;~&aring;&Eacute;&euml; E`~&euml;&Eacute; NOF
&ntilde;O &oacute;O
− =M
~ O &Auml;O
713. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml; m&auml;~&aring;&Eacute;&euml; E`~&euml;&Eacute; NPF
&ntilde;O &oacute;O
+ =M
~ O &Auml;O
714. m~&ecirc;~&Auml;&ccedil;&auml;&aacute;&Aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc; E`~&euml;&Eacute; NQF
&ntilde;O
−&oacute;=M
~O
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CHAPTER 7. ANALYTIC GEOMETRY
Figure 150.
715. o&Eacute;~&auml; m~&ecirc;~&auml;&auml;&Eacute;&auml; m&auml;~&aring;&Eacute;&euml; E`~&euml;&Eacute; NRF
&ntilde;O
=N
~O
716. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute; m~&ecirc;~&auml;&auml;&Eacute;&auml; m&auml;~&aring;&Eacute;&euml; E`~&euml;&Eacute; NSF
&ntilde;O
= −N
~O
717. `&ccedil;&aacute;&aring;&Aring;&aacute;&Ccedil;&Eacute;&aring;&iacute; m&auml;~&aring;&Eacute;&euml; E`~&euml;&Eacute; NTF
&ntilde;O = M
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CHAPTER 7. ANALYTIC GEOMETRY
7.12 Sphere
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; ~ &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W o
m&ccedil;&aacute;&aring;&iacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ntilde;I &oacute;I &ograve;I &ntilde; N I &oacute; N I &ograve; N I &pound;
`&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; ~ &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W (~ I &Auml;I &Aring; )
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ^I aI bI cI j
718. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ p&eacute;&Uuml;&Eacute;&ecirc;&Eacute; `&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil; ~&iacute; &iacute;&Uuml;&Eacute; l&ecirc;&aacute;&Ouml;&aacute;&aring; Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;
c&ccedil;&ecirc;&atilde;F
&ntilde; O + &oacute; O + &ograve; O = oO
Figure 151.
719. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ `&aacute;&ecirc;&Aring;&auml;&Eacute; `&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil; ~&iacute; ^&aring;&oacute; m&ccedil;&aacute;&aring;&iacute; (~ I &Auml;I &Aring; )
(&ntilde; − ~ )O + (&oacute; − &Auml;)O + (&ograve; − &Aring; )O = o O
720. a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc; c&ccedil;&ecirc;&atilde;
(&ntilde; − &ntilde; N )(&ntilde; − &ntilde; O ) + (&oacute; − &oacute; N )(&oacute; − &oacute; O ) + (&ograve; − &ograve;N )(&ograve; − &ograve; O ) = M I
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CHAPTER 7. ANALYTIC GEOMETRY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
mN (&ntilde; N I &oacute; N I &ograve; N ) I mO (&ntilde; O I &oacute; O I &ograve; O ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Eacute;&aring;&Ccedil;&euml; &ccedil;&Ntilde; ~ &Ccedil;&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;K
721. c&ccedil;&igrave;&ecirc; m&ccedil;&aacute;&aring;&iacute; c&ccedil;&ecirc;&atilde;
&ntilde;O + &oacute;O + &ograve;O &ntilde;
&ntilde; NO + &oacute; NO + &ntilde; NO &ntilde; N
&ntilde; OO + &oacute; OO + &ntilde; OO &ntilde; O
&ntilde; PO + &oacute; PO + &ntilde; PO &ntilde; P
&ntilde; OQ + &oacute; OQ + &ntilde; OQ &ntilde; Q
&oacute;
&ograve;
N
&oacute;N
&oacute;O
&ograve;N N
&ograve;O N = M
&oacute;P
&oacute;Q
&ograve;P N
&ograve;Q N
722. d&Eacute;&aring;&Eacute;&ecirc;~&auml; c&ccedil;&ecirc;&atilde;
^&ntilde; O + ^&oacute; O + ^&ograve; O + a&ntilde; + b&oacute; + c&ograve; + j = M E^ &aacute;&euml; &aring;&ccedil;&aring;&ograve;&Eacute;&ecirc;&ccedil;FK
q&Uuml;&Eacute; &Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute; &Uuml;~&euml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; (~ I &Auml;I &Aring; ) I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
a
b
c
~=−
I &Auml;=−
I &Aring;=−
K
O^
O^
O^
q&Uuml;&Eacute; &ecirc;~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute; &aacute;&euml;
aO + b O + c O − Q ^ O j
o=
K
O^
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Chapter 8
Differential Calculus
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde;I &Ouml;I &oacute;I &igrave;I &icirc;
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute; E&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;FW &ntilde;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W ~I &Auml;I &Aring;I &Ccedil;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
^&aring;&Ouml;&auml;&Eacute;W α
f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W &Ntilde; −N
8.1 Functions and Their Graphs
723. b&icirc;&Eacute;&aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Ntilde; (− &ntilde; ) = &Ntilde; (&ntilde; )
724. l&Ccedil;&Ccedil; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Ntilde; (− &ntilde; ) = −&Ntilde; (&ntilde; )
725. m&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Ntilde; (&ntilde; + &aring;q ) = &Ntilde; (&ntilde; )
726. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &Ntilde; (&ntilde; ) &aacute;&euml; ~&aring;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;I &ntilde; = &Ouml; (&oacute; ) &ccedil;&ecirc; &oacute; = &Ntilde; −N (&ntilde; ) &aacute;&euml; &aacute;&iacute;&euml; &aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;
&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K
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Figure 152.
727. `&ccedil;&atilde;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &Ntilde; (&igrave; ) I &igrave; = &Ouml; (&ntilde; ) I &oacute; = &Ntilde; (&Ouml; (&ntilde; )) &aacute;&euml; ~ &Aring;&ccedil;&atilde;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K
728. i&aacute;&aring;&Eacute;~&ecirc; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ntilde; + &Auml; I &ntilde; ∈ o I ~ = &iacute;~&aring; α &aacute;&euml; &iacute;&Uuml;&Eacute; &euml;&auml;&ccedil;&eacute;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;I &Auml; &aacute;&euml;
&iacute;&Uuml;&Eacute; &oacute;-&aacute;&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute;K
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Figure 153.
729. n&igrave;~&Ccedil;&ecirc;~&iacute;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &ntilde;O I &ntilde; ∈o K
Figure 154.
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CHAPTER 8. DIFFERENTIAL CALCULUS
730. &oacute; = ~&ntilde; O + &Auml;&ntilde; + &Aring; I &ntilde; ∈ o K
Figure 155.
731. `&igrave;&Auml;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &ntilde;P I &ntilde; ∈o K
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CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 156.
732. &oacute; = ~&ntilde; P + &Auml;&ntilde; O + &Aring;&ntilde; + &Ccedil; I &ntilde; ∈ o K
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Figure 157.
733. m&ccedil;&iuml;&Eacute;&ecirc; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &ntilde; &aring; I &aring;∈ k K
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Figure 158.
Figure 159.
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CHAPTER 8. DIFFERENTIAL CALCULUS
734. p&egrave;&igrave;~&ecirc;&Eacute; o&ccedil;&ccedil;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = &ntilde; I &ntilde; ∈ [MI ∞ ) K
Figure 160.
735. b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;
&oacute; = ~&ntilde; I ~ &gt; M I ~ ≠ NI
&oacute; = &Eacute; &ntilde; &aacute;&Ntilde; ~ = &Eacute; I &Eacute; = OKTNUOUNUOUQSK
Figure 161.
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CHAPTER 8. DIFFERENTIAL CALCULUS
736. i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&aacute;&Aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;
&oacute; = &auml;&ccedil;&Ouml; ~ &ntilde; I &ntilde; ∈ (MI ∞ ) I ~ &gt; M I ~ ≠ N I
&oacute; = &auml;&aring; &ntilde; &aacute;&Ntilde; ~ = &Eacute; I &ntilde; &gt; M K
Figure 162.
737. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; p&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Eacute;&ntilde; − &Eacute;−&ntilde;
I &ntilde; ∈o K
&oacute; = &euml;&aacute;&aring;&Uuml; &ntilde; I &euml;&aacute;&aring;&Uuml; &ntilde; =
O
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CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 163.
738. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&ccedil;&euml;&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Eacute;&ntilde; + &Eacute;−&ntilde;
I &ntilde; ∈o K
&oacute; = &Aring;&ccedil;&euml; &Uuml; &ntilde; I &Aring;&ccedil;&euml; &Uuml; &ntilde; =
O
Figure 164.
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CHAPTER 8. DIFFERENTIAL CALCULUS
739. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; q~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&euml;&aacute;&aring;&Uuml; &ntilde; &Eacute; &ntilde; − &Eacute; − &ntilde;
=
I &ntilde; ∈o K
&oacute; = &iacute;~&aring;&Uuml; &ntilde; I &oacute; = &iacute;~&aring;&Uuml; &ntilde; =
&Aring;&ccedil;&euml;&Uuml; &ntilde; &Eacute; &ntilde; + &Eacute; − &ntilde;
Figure 165.
740. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Aring;&ccedil;&euml;&Uuml; &ntilde; &Eacute; &ntilde; + &Eacute; − &ntilde;
=
I &ntilde; ∈o I &ntilde; ≠ M K
&oacute; = &Aring;&ccedil;&iacute; &Uuml; &ntilde; I &oacute; = &Aring;&ccedil;&iacute; &Uuml; &ntilde; =
&euml;&aacute;&aring;&Uuml; &ntilde; &Eacute; &ntilde; − &Eacute; − &ntilde;
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CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 166.
741. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; p&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
N
O
&oacute; = &euml;&Eacute;&Aring; &Uuml; &ntilde; I &oacute; = &euml;&Eacute;&Aring; &Uuml; &ntilde; =
= &ntilde; −&ntilde; I &ntilde; ∈ o K
&Aring;&ccedil;&euml;&Uuml; &ntilde; &Eacute; + &Eacute;
Figure 167.
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742. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
N
O
= &ntilde; −&ntilde; I &ntilde; ∈ o I &ntilde; ≠ M K
&oacute; = &Aring;&euml;&Aring;&Uuml; &ntilde; I &oacute; = &Aring;&euml;&Aring;&Uuml; &ntilde; =
&euml;&aacute;&aring;&Uuml; &ntilde; &Eacute; − &Eacute;
Figure 168.
743. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; p&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&euml;&aacute;&aring;&Uuml; &ntilde; I &ntilde; ∈ o K
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Figure 169.
744. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&ccedil;&euml;&aacute;&aring;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&Aring;&ccedil;&euml;&Uuml; &ntilde; I &ntilde; ∈ [NI ∞ ) K
Figure 170.
745. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; q~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&iacute;~&aring;&Uuml; &ntilde; I &ntilde; ∈ (− NI N) K
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CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 171.
746. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&Aring;&ccedil;&iacute;&Uuml; &ntilde; I &ntilde; ∈ (− ∞I − N) ∪ (NI ∞ ) K
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CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 172.
747. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; p&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&euml;&Eacute;&Aring;&Uuml; &ntilde; I &ntilde; ∈ (MI N] K
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CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 173.
748. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring; `&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&oacute; = ~&ecirc;&Aring;&Aring;&euml;&Aring;&Uuml; &ntilde; I &ntilde; ∈ o I &ntilde; ≠ M K
Figure 174.
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CHAPTER 8. DIFFERENTIAL CALCULUS
8.2 Limits of Functions
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde; (&ntilde; ) I &Ouml;(&ntilde; )
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;W &ntilde;
o&Eacute;~&auml; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;W ~I &acirc;
749. &auml;&aacute;&atilde;[&Ntilde; (&ntilde; ) + &Ouml;(&ntilde; )] = &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) + &auml;&aacute;&atilde; &Ouml; (&ntilde; )
&ntilde; →~
&ntilde; →~
&ntilde; →~
750. &auml;&aacute;&atilde;[&Ntilde; (&ntilde; ) − &Ouml; (&ntilde; )] = &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) − &auml;&aacute;&atilde; &Ouml; (&ntilde; )
&ntilde; →~
&ntilde; →~
&ntilde; →~
751. &auml;&aacute;&atilde;[&Ntilde; (&ntilde; ) ⋅ &Ouml; (&ntilde; )] = &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) ⋅ &auml;&aacute;&atilde; &Ouml; (&ntilde; )
&ntilde; →~
&ntilde; →~
&ntilde; →~
&Ntilde; (&ntilde; )
&Ntilde; (&ntilde; ) &auml;&aacute;&atilde;
= &ntilde; →~
I &aacute;&Ntilde; &auml;&aacute;&atilde; &Ouml; (&ntilde; ) ≠ M K
&ntilde; → ~ &Ouml; (&ntilde; )
&ntilde; →~
&auml;&aacute;&atilde; &Ouml; (&ntilde; )
752. &auml;&aacute;&atilde;
&ntilde; →~
753. &auml;&aacute;&atilde;[&acirc;&Ntilde; (&ntilde; )] = &acirc; &auml;&aacute;&atilde; &Ntilde; (&ntilde; )
&ntilde; →~
&ntilde; →~
(
)
754. &auml;&aacute;&atilde; &Ntilde; (&Ouml; (&ntilde; )) = &Ntilde; &auml;&aacute;&atilde; &Ouml; (&ntilde; )
&ntilde; →~
&ntilde; →~
755. &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) = &Ntilde; (~ ) I &aacute;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &Ntilde; (&ntilde; ) &aacute;&euml; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; ~&iacute; &ntilde; = ~ K
&ntilde; →~
&euml;&aacute;&aring; &ntilde;
=N
&ntilde; →M
&ntilde;
756. &auml;&aacute;&atilde;
757. &auml;&aacute;&atilde;
&ntilde; →M
&iacute;~&aring; &ntilde;
=N
&ntilde;
&euml;&aacute;&aring; −N &ntilde;
=N
&ntilde; →M
&ntilde;
758. &auml;&aacute;&atilde;
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CHAPTER 8. DIFFERENTIAL CALCULUS
&iacute;~&aring; −N &ntilde;
=N
&ntilde; →M
&ntilde;
759. &auml;&aacute;&atilde;
&auml;&aring;(N + &ntilde; )
=N
&ntilde; →M
&ntilde;
760. &auml;&aacute;&atilde;
&ntilde;
 N
761. &auml;&aacute;&atilde; N +  = &Eacute;
&ntilde; →∞
 &ntilde;
&ntilde;
 &acirc;
762. &auml;&aacute;&atilde; N +  = &Eacute; &acirc;
&ntilde; →∞
 &ntilde;
763. &auml;&aacute;&atilde; ~ &ntilde; = N
&ntilde; →M
8.3 Definition and Properties of the Derivative
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde;I &Ouml;I &oacute;I &igrave;I &icirc;
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W &ntilde;
o&Eacute;~&auml; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;W &acirc;
^&aring;&Ouml;&auml;&Eacute;W α
&Ntilde; (&ntilde; + ∆&ntilde; ) − &Ntilde; (&ntilde; )
∆&oacute; &Ccedil;&oacute;
= &auml;&aacute;&atilde;
=
∆&ntilde; → M
∆&ntilde; →M ∆&ntilde;
&Ccedil;&ntilde;
∆&ntilde;
764. &oacute;′(&ntilde; ) = &auml;&aacute;&atilde;
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Figure 175.
765.
&Ccedil;&oacute;
= &iacute;~&aring; α
&Ccedil;&ntilde;
766.
&Ccedil;(&igrave; + &icirc; ) &Ccedil;&igrave; &Ccedil;&icirc;
=
+
&Ccedil;&ntilde;
&Ccedil;&ntilde; &Ccedil;&ntilde;
767.
&Ccedil;(&igrave; − &icirc; ) &Ccedil;&igrave; &Ccedil;&icirc;
=
−
&Ccedil;&ntilde;
&Ccedil;&ntilde; &Ccedil;&ntilde;
768.
&Ccedil;(&acirc;&igrave; )
&Ccedil;&igrave;
=&acirc;
&Ccedil;&ntilde;
&Ccedil;&ntilde;
769. m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; o&igrave;&auml;&Eacute;
&Ccedil;(&igrave; ⋅ &icirc; ) &Ccedil;&igrave;
&Ccedil;&icirc;
=
⋅&icirc; + &igrave;⋅
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&ntilde;
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CHAPTER 8. DIFFERENTIAL CALCULUS
770. n&igrave;&ccedil;&iacute;&aacute;&Eacute;&aring;&iacute; o&igrave;&auml;&Eacute;
&Ccedil;&igrave;
&Ccedil;&icirc;
⋅&icirc; −&igrave;⋅
&Ccedil;  &igrave;  &Ccedil;&ntilde;
&Ccedil;&ntilde;
 =
O
&Ccedil;&ntilde;  &icirc; 
&icirc;
771. `&Uuml;~&aacute;&aring; o&igrave;&auml;&Eacute;
&oacute; = &Ntilde; (&Ouml;(&ntilde; )) I &igrave; = &Ouml;(&ntilde; ) I
&Ccedil;&oacute; &Ccedil;&oacute; &Ccedil;&igrave;
=
⋅
K
&Ccedil;&ntilde; &Ccedil;&igrave; &Ccedil;&ntilde;
772. a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; &ccedil;&Ntilde; f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Ccedil;&oacute;
N
=
I
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&oacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &ntilde; (&oacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &oacute; (&ntilde; ) K
773. o&Eacute;&Aring;&aacute;&eacute;&ecirc;&ccedil;&Aring;~&auml; o&igrave;&auml;&Eacute;
&Ccedil;&oacute;
&Ccedil;  N
  = − &Ccedil;&ntilde;O
&Ccedil;&ntilde;  &oacute; 
&oacute;
774. i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&aacute;&Aring; a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&iacute;&aacute;&ccedil;&aring;
&oacute; = &Ntilde; (&ntilde; ) I &auml;&aring; &oacute; = &auml;&aring; &Ntilde; (&ntilde; ) I
&Ccedil;&oacute;
&Ccedil;
= &Ntilde; (&ntilde; ) ⋅ [&auml;&aring; &Ntilde; (&ntilde; )] K
&Ccedil;&ntilde;
&Ccedil;&ntilde;
8.4 Table of Derivatives
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W &ntilde;
o&Eacute;~&auml; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;W `I ~I &Auml;I &Aring;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
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775.
&Ccedil;
(` ) = M
&Ccedil;&ntilde;
776.
&Ccedil;
(&ntilde; ) = N
&Ccedil;&ntilde;
777.
&Ccedil;
(~&ntilde; + &Auml;) = ~
&Ccedil;&ntilde;
778.
&Ccedil;
(~&ntilde; O + &Auml;&ntilde; + &Aring; ) = ~&ntilde; + &Auml;
&Ccedil;&ntilde;
779.
&Ccedil; &aring;
(
&ntilde; ) = &aring;&ntilde; &aring; −N
&Ccedil;&ntilde;
780.
&Ccedil; −&aring;
&aring;
(
&ntilde; ) = − &aring; +N
&ntilde;
&Ccedil;&ntilde;
781.
&Ccedil; N
N
 =− O
&Ccedil;&ntilde;  &ntilde; 
&ntilde;
782.
&Ccedil;
&Ccedil;&ntilde;
( &ntilde; ) = O N&ntilde;
783.
&Ccedil;
&Ccedil;&ntilde;
( &ntilde; )=
784.
&Ccedil;
(&auml;&aring; &ntilde; ) = N
&ntilde;
&Ccedil;&ntilde;
785.
&Ccedil;
(&auml;&ccedil;&Ouml; ~ &ntilde; ) = N I ~ &gt; M I ~ ≠ N K
&ntilde; &auml;&aring; ~
&Ccedil;&ntilde;
N
&aring;
&aring; &aring; &ntilde; &aring; −N
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786.
&Ccedil; &ntilde;
(
~ ) = ~ &ntilde; &auml;&aring; ~ I ~ &gt; M I ~ ≠ N K
&Ccedil;&ntilde;
787.
&Ccedil; &ntilde;
(
&Eacute; ) = &Eacute;&ntilde;
&Ccedil;&ntilde;
788.
&Ccedil;
(&euml;&aacute;&aring; &ntilde; ) = &Aring;&ccedil;&euml; &ntilde;
&Ccedil;&ntilde;
789.
&Ccedil;
(&Aring;&ccedil;&euml; &ntilde; ) = − &euml;&aacute;&aring; &ntilde;
&Ccedil;&ntilde;
790.
&Ccedil;
(&iacute;~&aring; &ntilde; ) = NO = &euml;&Eacute;&Aring; O &ntilde;
&Aring;&ccedil;&euml; &ntilde;
&Ccedil;&ntilde;
791.
&Ccedil;
(&Aring;&ccedil;&iacute; &ntilde; ) = − NO = − &Aring;&euml;&Aring; O &ntilde;
&euml;&aacute;&aring; &ntilde;
&Ccedil;&ntilde;
792.
&Ccedil;
(&euml;&Eacute;&Aring; &ntilde; ) = &iacute;~&aring; &ntilde; ⋅ &euml;&Eacute;&Aring; &ntilde;
&Ccedil;&ntilde;
793.
&Ccedil;
(&Aring;&euml;&Aring; &ntilde; ) = − &Aring;&ccedil;&iacute; &ntilde; ⋅ &Aring;&euml;&Aring; &ntilde;
&Ccedil;&ntilde;
794.
&Ccedil;
(~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; ) = N O
&Ccedil;&ntilde;
N− &ntilde;
795.
&Ccedil;
(~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; ) = − N O
&Ccedil;&ntilde;
N− &ntilde;
796.
&Ccedil;
(~&ecirc;&Aring;&iacute;~&aring; &ntilde; ) = N O
N+ &ntilde;
&Ccedil;&ntilde;
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797.
&Ccedil;
(~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; ) = − N O
N+ &ntilde;
&Ccedil;&ntilde;
798.
&Ccedil;
(~&ecirc;&Aring; &euml;&Eacute;&Aring; &ntilde; ) = NO
&Ccedil;&ntilde;
&ntilde; &ntilde; −N
799.
&Ccedil;
(~&ecirc;&Aring; &Aring;&euml;&Aring; &ntilde; ) = − NO
&Ccedil;&ntilde;
&ntilde; &ntilde; −N
800.
&Ccedil;
(&euml;&aacute;&aring;&Uuml; &ntilde; ) = &Aring;&ccedil;&euml;&Uuml; &ntilde;
&Ccedil;&ntilde;
801.
&Ccedil;
(&Aring;&ccedil;&euml;&Uuml; &ntilde; ) = &euml;&aacute;&aring;&Uuml; &ntilde;
&Ccedil;&ntilde;
802.
&Ccedil;
(&iacute;~&aring;&Uuml; &ntilde; ) = N O = &euml;&Eacute;&Aring;&Uuml; O &ntilde;
&Aring;&ccedil;&euml;&Uuml; &ntilde;
&Ccedil;&ntilde;
803.
&Ccedil;
(&Aring;&ccedil;&iacute;&Uuml; &ntilde; ) = − N O = −&Aring;&euml;&Aring;&Uuml; O &ntilde;
&euml;&aacute;&aring;&Uuml; &ntilde;
&Ccedil;&ntilde;
804.
&Ccedil;
(&euml;&Eacute;&Aring;&Uuml; &ntilde; ) = −&euml;&Eacute;&Aring;&Uuml; &ntilde; ⋅ &iacute;~&aring;&Uuml; &ntilde;
&Ccedil;&ntilde;
805.
&Ccedil;
(&Aring;&euml;&Aring;&Uuml; &ntilde; ) = −&Aring;&euml;&Aring;&Uuml; &ntilde; ⋅ &Aring;&ccedil;&iacute;&Uuml; &ntilde;
&Ccedil;&ntilde;
806.
&Ccedil;
(~&ecirc;&Aring;&euml;&aacute;&aring;&Uuml; &ntilde; ) = NO
&Ccedil;&ntilde;
&ntilde; +N
807.
&Ccedil;
(~&ecirc;&Aring;&Aring;&ccedil;&euml;&Uuml; &ntilde; ) = ON
&Ccedil;&ntilde;
&ntilde; −N
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808.
&Ccedil;
(~&ecirc;&Aring;&iacute;~&aring;&Uuml; &ntilde; ) = N O I &ntilde; &lt; N K
&Ccedil;&ntilde;
N− &ntilde;
809.
&Ccedil;
(~&ecirc;&Aring;&Aring;&ccedil;&iacute;&Uuml; &ntilde; ) = − ON I &ntilde; &gt; N K
&Ccedil;&ntilde;
&ntilde; −N
810.
&Ccedil; &icirc;
&Ccedil;&igrave;
&Ccedil;&icirc;
(
+ &igrave; &icirc; &auml;&aring; &igrave; ⋅
&igrave; ) = &icirc;&igrave; &icirc; −N ⋅
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&ntilde;
8.5 Higher Order Derivatives
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde;I &oacute;I &igrave;I &icirc;
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W &ntilde;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
811. p&Eacute;&Aring;&ccedil;&aring;&Ccedil; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;
O
 &Ccedil;&oacute;  ′ &Ccedil;  &Ccedil;&oacute;  &Ccedil; &oacute;
′
&Ntilde; ′′ = (&Ntilde; ′) =   =
 = O
 &Ccedil;&ntilde;  &Ccedil;&ntilde;  &Ccedil;&ntilde;  &Ccedil;&ntilde;
812. e&aacute;&Ouml;&Uuml;&Eacute;&ecirc;-l&ecirc;&Ccedil;&Eacute;&ecirc; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;
&Ccedil;&aring; &oacute;
(&aring; )
&Ntilde; = &aring; = &oacute; (&aring; ) = (&Ntilde; (&aring; −N) ) ′
&Ccedil;&ntilde;
813.
(&igrave; + &icirc; )(&aring; ) = &igrave; (&aring; ) + &icirc; (&aring; )
814.
(&igrave; − &icirc; )(&aring; ) = &igrave;(&aring; ) − &icirc; (&aring; )
815. i&Eacute;&aacute;&Auml;&aring;&aacute;&iacute;&ograve;∞&euml; c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;
(&igrave;&icirc; )′′ = &igrave;′′&icirc; + O&igrave;′&icirc;′ + &igrave;&icirc;′′
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(&igrave;&icirc; )′′′ = &igrave;′′′&icirc; + P&igrave;′′&icirc;′ + P&igrave;′&icirc;′′ + &igrave;&icirc;′′′
(&igrave;&icirc; )(&aring; ) = &igrave; (&aring; ) &icirc; + &aring;&igrave; (&aring;−N) &icirc; ′ + &aring;(&aring; − N) &igrave; (&aring;−O) &icirc; ′′ + K + &igrave;&icirc; (&aring; )
N⋅ O
816.
(&ntilde; )( ) = (&atilde;&atilde;−&gt;&aring;)&gt; &ntilde;
817.
(&ntilde; )( ) = &aring;&gt;
818.
&aring; −N
(
− N) (&aring; − N)&gt;
(&auml;&ccedil;&Ouml; ~ &ntilde; ) =
&aring;
819.
&aring; −N
(&auml;&aring; &ntilde; )(&aring; ) = (− N) (&aring;&aring; − N)&gt;
820.
(~ )( ) = ~
&aring;
&ntilde;
821.
(&Eacute; )( ) = &Eacute;
&ntilde;
822.
(~ )( ) = &atilde; ~
823.
(&euml;&aacute;&aring; &ntilde; )(&aring; ) = &euml;&aacute;&aring; &ntilde; + &aring;π 
824.
(&Aring;&ccedil;&euml; &ntilde; )(&aring; ) = &Aring;&ccedil;&euml; &ntilde; + &aring;π 
&atilde;
&aring;
&aring;
&atilde; −&aring;
&aring;
(&aring; )
&ntilde; &auml;&aring; ~
&ntilde;
&ntilde;
&ntilde;
&atilde;&ntilde;
&aring;
&aring;
&auml;&aring; &aring; ~
&aring; &atilde;&ntilde;


&auml;&aring; &aring; ~
O 
O 
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CHAPTER 8. DIFFERENTIAL CALCULUS
8.6 Applications of Derivative
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde;I &Ouml;I &oacute;
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~&aring; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;W &euml;
s&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;W &icirc;
^&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;W &iuml;
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W &ntilde;
q&aacute;&atilde;&Eacute;W &iacute;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
825. s&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute; ~&aring;&Ccedil; ^&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;
&euml; = &Ntilde; (&iacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~&aring; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute; &ecirc;&Eacute;&auml;~&iacute;&aacute;&icirc;&Eacute; &iacute;&ccedil; ~ &Ntilde;&aacute;&ntilde;&Eacute;&Ccedil;
&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; &euml;&oacute;&euml;&iacute;&Eacute;&atilde; ~&iacute; ~ &iacute;&aacute;&atilde;&Eacute; &iacute;I
&icirc; = &euml;′ = &Ntilde; ′(&iacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&euml;&iacute;~&aring;&iacute;~&aring;&Eacute;&ccedil;&igrave;&euml; &icirc;&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;I
&iuml; = &icirc;′ = &euml;′′ = &Ntilde; ′′(&iacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&euml;&iacute;~&aring;&iacute;~&aring;&Eacute;&ccedil;&igrave;&euml; ~&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde;
&iacute;&Uuml;&Eacute; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;K
826. q~&aring;&Ouml;&Eacute;&aring;&iacute; i&aacute;&aring;&Eacute;
&oacute; − &oacute; M = &Ntilde; ′(&ntilde; M )(&ntilde; − &ntilde; M )
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CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 176.
827. k&ccedil;&ecirc;&atilde;~&auml; i&aacute;&aring;&Eacute;
N
(&ntilde; − &ntilde; M ) Ec&aacute;&Ouml; NTSF
&oacute; − &oacute;M = −
&Ntilde; ′(&ntilde; M )
828. f&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; ~&aring;&Ccedil; a&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;K
f&Ntilde; &Ntilde; ′(&ntilde; M ) &gt; M I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &aacute;&euml; &aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; ~&iacute; &ntilde; M K Ec&aacute;&Ouml; NTTI &ntilde; &lt; &ntilde; N I
&ntilde; O &lt; &ntilde; FI
f&Ntilde; &Ntilde; ′(&ntilde; M ) &lt; M I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &aacute;&euml; &Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; ~&iacute; &ntilde; M K Ec&aacute;&Ouml; NTTI
&ntilde; N &lt; &ntilde; &lt; &ntilde; O FI
f&Ntilde; &Ntilde; ′ (&ntilde; M ) &Ccedil;&ccedil;&Eacute;&euml; &aring;&ccedil;&iacute; &Eacute;&ntilde;&aacute;&euml;&iacute; &ccedil;&ecirc; &aacute;&euml; &ograve;&Eacute;&ecirc;&ccedil;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &iacute;&Eacute;&euml;&iacute; &Ntilde;~&aacute;&auml;&euml;K
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Figure 177.
829. i&ccedil;&Aring;~&auml; &Eacute;&ntilde;&iacute;&ecirc;&Eacute;&atilde;~
^ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &Ntilde;E&ntilde;F &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde; ~&iacute; &ntilde; N &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde;
&iacute;&Uuml;&Eacute;&ecirc;&Eacute; &Eacute;&ntilde;&aacute;&euml;&iacute;&euml; &euml;&ccedil;&atilde;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; &Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&aacute;&aring;&Ouml; &ntilde; N &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute;
&Ntilde; (&ntilde; N ) ≥ &Ntilde; (&ntilde; ) &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; &aacute;&aring; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; Ec&aacute;&Ouml;KNTTFK
^ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &Ntilde;E&ntilde;F &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde; ~&iacute; &ntilde; O &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde;
&iacute;&Uuml;&Eacute;&ecirc;&Eacute; &Eacute;&ntilde;&aacute;&euml;&iacute;&euml; &euml;&ccedil;&atilde;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; &Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&aacute;&aring;&Ouml; &ntilde; O &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute;
&Ntilde; (&ntilde; O ) ≤ &Ntilde; (&ntilde; ) &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; &aacute;&aring; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; Ec&aacute;&Ouml;KNTTFK
830. `&ecirc;&aacute;&iacute;&aacute;&Aring;~&auml; m&ccedil;&aacute;&aring;&iacute;&euml;
^ &Aring;&ecirc;&aacute;&iacute;&aacute;&Aring;~&auml; &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&aring; &Ntilde;E&ntilde;F &ccedil;&Aring;&Aring;&igrave;&ecirc;&euml; ~&iacute; &ntilde; M &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde; &Eacute;&aacute;&iacute;&Uuml;&Eacute;&ecirc;
&Ntilde; ′ (&ntilde; M ) &aacute;&euml; &ograve;&Eacute;&ecirc;&ccedil; &ccedil;&ecirc; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; &Ccedil;&ccedil;&Eacute;&euml;&aring;∞&iacute; &Eacute;&ntilde;&aacute;&euml;&iacute;K
831. c&aacute;&ecirc;&euml;&iacute; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; q&Eacute;&euml;&iacute; &Ntilde;&ccedil;&ecirc; i&ccedil;&Aring;~&auml; b&ntilde;&iacute;&ecirc;&Eacute;&atilde;~K
f&Ntilde; &Ntilde;E&ntilde;F &aacute;&euml; &aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; E &Ntilde; ′ (&ntilde; ) &gt; M F &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; &aacute;&aring; &euml;&ccedil;&atilde;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;
(~I &ntilde; N ] ~&aring;&Ccedil; &Ntilde;E&ntilde;F &aacute;&euml; &Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; E &Ntilde; ′ (&ntilde; ) &lt; M F &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; &aacute;&aring; &euml;&ccedil;&atilde;&Eacute;
&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; [&ntilde; N I &Auml;) I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde; ~&iacute; &ntilde; N
Ec&aacute;&Ouml;KNTTFK
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832. f&Ntilde; &Ntilde;E&ntilde;F &aacute;&euml; &Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; E &Ntilde; ′ (&ntilde; ) &lt; M F &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; &aacute;&aring; &euml;&ccedil;&atilde;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;
(~I &ntilde; O ] ~&aring;&Ccedil; &Ntilde;E&ntilde;F &aacute;&euml; &aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; E &Ntilde; ′ (&ntilde; ) &gt; M F &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; &aacute;&aring; &euml;&ccedil;&atilde;&Eacute;
&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; [&ntilde; O I &Auml;) I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde; ~&iacute; &ntilde; O K
Ec&aacute;&Ouml;KNTTFK
833. p&Eacute;&Aring;&ccedil;&aring;&Ccedil; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; q&Eacute;&euml;&iacute; &Ntilde;&ccedil;&ecirc; i&ccedil;&Aring;~&auml; b&ntilde;&iacute;&ecirc;&Eacute;&atilde;~K
f&Ntilde; &Ntilde; ′ (&ntilde; N ) = M ~&aring;&Ccedil; &Ntilde; ′′(&ntilde; N ) &lt; M I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde;
~&iacute; &ntilde; N K
f&Ntilde; &Ntilde; ′ (&ntilde; O ) = M ~&aring;&Ccedil; &Ntilde; ′′(&ntilde; O ) &gt; M I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde;
~&iacute; &ntilde; O K Ec&aacute;&Ouml;KNTTF
834. `&ccedil;&aring;&Aring;~&icirc;&aacute;&iacute;&oacute;K
&Ntilde;E&ntilde;F &aacute;&euml; &Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute; &igrave;&eacute;&iuml;~&ecirc;&Ccedil; ~&iacute; &ntilde; M &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde; &Ntilde; ′ (&ntilde; ) &aacute;&euml;
&aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; ~&iacute; &ntilde; M Ec&aacute;&Ouml;KNTTI &ntilde; P &lt; &ntilde; FK
&Ntilde;E&ntilde;F &aacute;&euml; &Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute; &Ccedil;&ccedil;&iuml;&aring;&iuml;~&ecirc;&Ccedil; ~&iacute; &ntilde; M &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde; &Ntilde; ′ (&ntilde; ) &aacute;&euml;
&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; ~&iacute; &ntilde; M K Ec&aacute;&Ouml;KNTTI &ntilde; &lt; &ntilde; P FK
835. p&Eacute;&Aring;&ccedil;&aring;&Ccedil; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; q&Eacute;&euml;&iacute; &Ntilde;&ccedil;&ecirc; `&ccedil;&aring;&Aring;~&icirc;&aacute;&iacute;&oacute;K
f&Ntilde; &Ntilde; ′′(&ntilde; M ) &gt; M I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &aacute;&euml; &Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute; &igrave;&eacute;&iuml;~&ecirc;&Ccedil; ~&iacute; &ntilde; M K
f&Ntilde; &Ntilde; ′′(&ntilde; M ) &lt; M I &iacute;&Uuml;&Eacute;&aring; &Ntilde;E&ntilde;F &aacute;&euml; &Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute; &Ccedil;&ccedil;&iuml;&aring;&iuml;~&ecirc;&Ccedil; ~&iacute; &ntilde; M K
f&Ntilde; &Ntilde; ′′(&ntilde; ) &Ccedil;&ccedil;&Eacute;&euml; &aring;&ccedil;&iacute; &Eacute;&ntilde;&aacute;&euml;&iacute; &ccedil;&ecirc; &aacute;&euml; &ograve;&Eacute;&ecirc;&ccedil;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &iacute;&Eacute;&euml;&iacute; &Ntilde;~&aacute;&auml;&euml;K
836. f&aring;&Ntilde;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; m&ccedil;&aacute;&aring;&iacute;&euml;
f&Ntilde; &Ntilde; ′ (&ntilde; P ) &Eacute;&ntilde;&aacute;&euml;&iacute;&euml; ~&aring;&Ccedil; &Ntilde; ′′(&ntilde; ) &Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml; &euml;&aacute;&Ouml;&aring; ~&iacute; &ntilde; = &ntilde; P I &iacute;&Uuml;&Eacute;&aring;
&iacute;&Uuml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; (&ntilde; P I &Ntilde; (&ntilde; P )) &aacute;&euml; ~&aring; &aacute;&aring;&Ntilde;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ouml;&ecirc;~&eacute;&Uuml; &ccedil;&Ntilde;
&Ntilde; (&ntilde; ) K f&Ntilde; &Ntilde; ′′(&ntilde; P ) &Eacute;&ntilde;&aacute;&euml;&iacute;&euml; ~&iacute; &iacute;&Uuml;&Eacute; &aacute;&aring;&Ntilde;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &eacute;&ccedil;&aacute;&aring;&iacute;I &iacute;&Uuml;&Eacute;&aring; &Ntilde; ′′(&ntilde; P ) = M
Ec&aacute;&Ouml;KNTTFK
837. i∞e&ccedil;&eacute;&aacute;&iacute;~&auml;∞&euml; o&igrave;&auml;&Eacute;
M
&Ntilde; ′(&ntilde; )
&Ntilde; (&ntilde; )
= &auml;&aacute;&atilde;
&aacute;&Ntilde; &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) = &auml;&aacute;&atilde; &Ouml; (&ntilde; ) =  K
&auml;&aacute;&atilde;
&ntilde; → &Aring; &Ouml; (&ntilde; )
&ntilde; →&Aring; &Ouml; ′(&ntilde; )
&ntilde; →&Aring;
&ntilde; →&Aring;
∞
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8.7 Differential
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde;I &igrave;I &icirc;
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W &ntilde;
a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; &ccedil;&Ntilde; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W &oacute; ′(&ntilde; ) I &Ntilde; ′(&ntilde; )
o&Eacute;~&auml; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;W `
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &ccedil;&Ntilde; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &oacute; = &Ntilde; (&ntilde; ) W &Ccedil;&oacute;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &ccedil;&Ntilde; &ntilde;W &Ccedil;&ntilde;
p&atilde;~&auml;&auml; &Aring;&Uuml;~&aring;&Ouml;&Eacute; &aacute;&aring; &ntilde;W ∆&ntilde;
p&atilde;~&auml;&auml; &Aring;&Uuml;~&aring;&Ouml;&Eacute; &aacute;&aring; &oacute;W ∆&oacute;
838. &Ccedil;&oacute; = &oacute; ′&Ccedil;&ntilde;
839. &Ntilde; (&ntilde; + ∆&ntilde; ) = &Ntilde; (&ntilde; ) + &Ntilde; ′(&ntilde; )∆&ntilde;
Figure 178.
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CHAPTER 8. DIFFERENTIAL CALCULUS
840. p&atilde;~&auml;&auml; `&Uuml;~&aring;&Ouml;&Eacute; &aacute;&aring; &oacute;
∆&oacute; = &Ntilde; (&ntilde; + ∆&ntilde; ) − &Ntilde; (&ntilde; )
841. &Ccedil;(&igrave; + &icirc; ) = &Ccedil;&igrave; + &Ccedil;&icirc;
842. &Ccedil;(&igrave; − &icirc; ) = &Ccedil;&igrave; − &Ccedil;&icirc;
843. &Ccedil;(`&igrave; ) = `&Ccedil;&igrave;
844. &Ccedil;(&igrave;&icirc; ) = &icirc;&Ccedil;&igrave; + &igrave;&Ccedil;&icirc;
 &igrave;  &icirc;&Ccedil;&igrave; − &igrave;&Ccedil;&icirc;
845. &Ccedil;  =
&icirc;O
&icirc;
8.8 Multivariable Functions
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&iuml;&ccedil; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &ograve; (&ntilde; I &oacute; ) I &Ntilde; (&ntilde; I &oacute; ) I &Ouml; (&ntilde; I &oacute; ) I &Uuml;(&ntilde; I &oacute; )
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;&euml;W &ntilde;I &oacute;I &iacute;
p&atilde;~&auml;&auml; &Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml; &aacute;&aring; &ntilde;I &oacute;I &ograve;I &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;W ∆&ntilde; I ∆&oacute; I ∆&ograve; K
846. c&aacute;&ecirc;&euml;&iacute; l&ecirc;&Ccedil;&Eacute;&ecirc; m~&ecirc;&iacute;&aacute;~&auml; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;
q&Uuml;&Eacute; &eacute;~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; &iuml;&aacute;&iacute;&Uuml; &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute; &iacute;&ccedil; &ntilde;
∂&Ntilde;
∂&ograve;
= &Ntilde; &ntilde; E~&auml;&euml;&ccedil;
= &ograve; &ntilde; FI
∂&ntilde;
∂&ntilde;
q&Uuml;&Eacute; &eacute;~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; &iuml;&aacute;&iacute;&Uuml; &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute; &iacute;&ccedil; &oacute;
∂&Ntilde;
∂&ograve;
= &Ntilde; &oacute; E~&auml;&euml;&ccedil;
= &ograve; &oacute; FK
∂&oacute;
∂&oacute;
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CHAPTER 8. DIFFERENTIAL CALCULUS
847. p&Eacute;&Aring;&ccedil;&aring;&Ccedil; l&ecirc;&Ccedil;&Eacute;&ecirc; m~&ecirc;&iacute;&aacute;~&auml; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;
∂  ∂&Ntilde;  ∂ O&Ntilde;
= &Ntilde; &ntilde;&ntilde; I
 =
∂&ntilde;  ∂&ntilde;  ∂&ntilde; O
∂  ∂&Ntilde;  ∂ O&Ntilde;
 =
= &Ntilde; &oacute;&oacute; I
∂&oacute;  ∂&oacute;  ∂&oacute; O
∂  ∂&Ntilde;  ∂ O&Ntilde;
= &Ntilde; &ntilde;&oacute; I
 =
∂&oacute;  ∂&ntilde;  ∂&oacute;∂&ntilde;
∂  ∂&Ntilde;  ∂ O&Ntilde;
 =
= &Ntilde; &oacute;&ntilde; K
∂&ntilde;  ∂&oacute;  ∂&ntilde;∂&oacute;
f&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml; ~&ecirc;&Eacute; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;I &iacute;&Uuml;&Eacute;&aring;
∂ O&Ntilde;
∂ O&Ntilde;
=
K
∂&oacute;∂&ntilde; ∂&ntilde;∂&oacute;
848. `&Uuml;~&aacute;&aring; o&igrave;&auml;&Eacute;&euml;
f&Ntilde; &Ntilde; (&ntilde; I &oacute; ) = &Ouml;(&Uuml;(&ntilde; I &oacute; )) E&Ouml; &aacute;&euml; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &ccedil;&aring;&Eacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute; &Uuml;FI &iacute;&Uuml;&Eacute;&aring;
∂&Uuml; ∂&Ntilde;
∂&Uuml;
∂&Ntilde;
= &Ouml;′(&Uuml;(&ntilde; I &oacute; )) I
= &Ouml;′(&Uuml;(&ntilde; I &oacute; )) K
∂&ntilde; ∂&oacute;
∂&ntilde;
∂&oacute;
f&Ntilde; &Uuml;(&iacute; ) = &Ntilde; (&ntilde; (&iacute; )I &oacute; (&iacute; )) I &iacute;&Uuml;&Eacute;&aring; &Uuml;′(&iacute; ) =
∂&Ntilde; &Ccedil;&ntilde; ∂&Ntilde; &Ccedil;&oacute;
+
K
∂&ntilde; &Ccedil;&iacute; ∂&oacute; &Ccedil;&iacute;
f&Ntilde; &ograve; = &Ntilde; (&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )) I &iacute;&Uuml;&Eacute;&aring;
∂&ograve; ∂&Ntilde; ∂&ntilde; ∂&Ntilde; ∂&oacute; ∂&ograve; ∂&Ntilde; ∂&ntilde; ∂&Ntilde; ∂&oacute;
=
+
K
+
=
I
∂&igrave; ∂&ntilde; ∂&igrave; ∂&oacute; ∂&igrave; ∂&icirc; ∂&ntilde; ∂&icirc; ∂&oacute; ∂&icirc;
849. p&atilde;~&auml;&auml; `&Uuml;~&aring;&Ouml;&Eacute;&euml;
∂&Ntilde;
∂&Ntilde;
∆&ograve; ≈ ∆&ntilde; + ∆&oacute;
∂&oacute;
∂&ntilde;
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CHAPTER 8. DIFFERENTIAL CALCULUS
850. i&ccedil;&Aring;~&auml; j~&ntilde;&aacute;&atilde;~ ~&aring;&Ccedil; j&aacute;&aring;&aacute;&atilde;~
&Ntilde; (&ntilde; I &oacute; ) &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde; ~&iacute; (&ntilde; M I &oacute; M ) &aacute;&Ntilde; &Ntilde; (&ntilde; I &oacute; ) ≤ &Ntilde; (&ntilde; M I &oacute; M )
&Ntilde;&ccedil;&ecirc; ~&auml;&auml; (&ntilde; I &oacute; ) &euml;&igrave;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&auml;&oacute; &Aring;&auml;&ccedil;&euml;&Eacute; &iacute;&ccedil; (&ntilde; M I &oacute; M ) K
&Ntilde; (&ntilde; I &oacute; ) &Uuml;~&euml; ~ &auml;&ccedil;&Aring;~&auml; &atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde; ~&iacute; (&ntilde; M I &oacute; M ) &aacute;&Ntilde; &Ntilde; (&ntilde; I &oacute; ) ≥ &Ntilde; (&ntilde; M I &oacute; M )
&Ntilde;&ccedil;&ecirc; ~&auml;&auml; (&ntilde; I &oacute; ) &euml;&igrave;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&auml;&oacute; &Aring;&auml;&ccedil;&euml;&Eacute; &iacute;&ccedil; (&ntilde; M I &oacute; M ) K
851. p&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute; m&ccedil;&aacute;&aring;&iacute;&euml;
∂&Ntilde; ∂&Ntilde;
=MK
=
∂&ntilde; ∂&oacute;
i&ccedil;&Aring;~&auml; &atilde;~&ntilde;&aacute;&atilde;~ ~&aring;&Ccedil; &auml;&ccedil;&Aring;~&auml; &atilde;&aacute;&aring;&aacute;&atilde;~ &ccedil;&Aring;&Aring;&igrave;&ecirc; ~&iacute; &euml;&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute; &eacute;&ccedil;&aacute;&aring;&iacute;&euml;K
852. p~&Ccedil;&Ccedil;&auml;&Eacute; m&ccedil;&aacute;&aring;&iacute;
^ &euml;&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute; &eacute;&ccedil;&aacute;&aring;&iacute; &iuml;&Uuml;&aacute;&Aring;&Uuml; &aacute;&euml; &aring;&Eacute;&aacute;&iacute;&Uuml;&Eacute;&ecirc; ~ &auml;&ccedil;&Aring;~&auml; &atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde;
&aring;&ccedil;&ecirc; ~ &auml;&ccedil;&Aring;~&auml; &atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde;
853. p&Eacute;&Aring;&ccedil;&aring;&Ccedil; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; q&Eacute;&euml;&iacute; &Ntilde;&ccedil;&ecirc; p&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute; m&ccedil;&aacute;&aring;&iacute;&euml;
∂&Ntilde; ∂&Ntilde;
i&Eacute;&iacute; (&ntilde; M I &oacute; M ) &Auml;&Eacute; ~ &euml;&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute; &eacute;&ccedil;&aacute;&aring;&iacute; E
=
= M FK
∂&ntilde; ∂&oacute;
&Ntilde; &ntilde;&ntilde; (&ntilde; M I &oacute; M ) &Ntilde; &ntilde;&oacute; (&ntilde; M I &oacute; M )
K
a=
&Ntilde; &oacute;&ntilde; (&ntilde; M I &oacute; M ) &Ntilde; &oacute;&oacute; (&ntilde; M I &oacute; M )
f&Ntilde;
f&Ntilde;
f&Ntilde;
f&Ntilde;
a &gt; M I &Ntilde; &ntilde;&ntilde; (&ntilde; M I &oacute; M ) &gt; M I (&ntilde; M I &oacute; M ) &aacute;&euml; ~ &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; &auml;&ccedil;&Aring;~&auml; &atilde;&aacute;&aring;&aacute;&atilde;~K
a &gt; M I &Ntilde; &ntilde;&ntilde; (&ntilde; M I &oacute; M ) &lt; M I (&ntilde; M I &oacute; M ) &aacute;&euml; ~ &eacute;&ccedil;&aacute;&aring;&iacute; &ccedil;&Ntilde; &auml;&ccedil;&Aring;~&auml; &atilde;~&ntilde;&aacute;&atilde;~K
a &lt; M I (&ntilde; M I &oacute; M ) &aacute;&euml; ~ &euml;~&Ccedil;&Ccedil;&auml;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute;K
a = M I &iacute;&Uuml;&Eacute; &iacute;&Eacute;&euml;&iacute; &Ntilde;~&aacute;&auml;&euml;K
854. q~&aring;&Ouml;&Eacute;&aring;&iacute; m&auml;~&aring;&Eacute;
q&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &iacute;~&aring;&Ouml;&Eacute;&aring;&iacute; &eacute;&auml;~&aring;&Eacute; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &ograve; = &Ntilde; (&ntilde; I &oacute; )
~&iacute; (&ntilde; M I &oacute; M I &ograve; M ) &aacute;&euml;
&ograve; − &ograve; M = &Ntilde; &ntilde; (&ntilde; M I &oacute; M )(&ntilde; − &ntilde; M ) + &Ntilde; &oacute; (&ntilde; M I &oacute; M )(&oacute; − &oacute; M ) K
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855. k&ccedil;&ecirc;&atilde;~&auml; &iacute;&ccedil; p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;
q&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &aring;&ccedil;&ecirc;&atilde;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &ograve; = &Ntilde; (&ntilde; I &oacute; ) ~&iacute;
(&ntilde; M I &oacute; M I &ograve; M ) &aacute;&euml;
&ntilde; − &ntilde;M
&oacute; − &oacute;M
&ograve; − &ograve;M
=
=
K
&Ntilde; &ntilde; ( &ntilde; M I &oacute; M ) &Ntilde; &oacute; (&ntilde; M I &oacute; M )
−N
8.9 Differential Operators
r r r
r&aring;&aacute;&iacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml; ~&auml;&ccedil;&aring;&Ouml; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; ~&ntilde;&Eacute;&euml;W &aacute; I &agrave; I &acirc;
p&Aring;~&auml;~&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; E&euml;&Aring;~&auml;~&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;&euml;FW &Ntilde; (&ntilde; I &oacute; I &ograve; ) I &igrave;(&ntilde; N I &ntilde; O I KI &ntilde; &aring; )
d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute; &ccedil;&Ntilde; ~ &euml;&Aring;~&auml;~&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W &Ouml;&ecirc;~&Ccedil; &igrave; I ∇&igrave;
∂&Ntilde;
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;W
∂&auml;
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; E&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;FW c (mI nI o )
r
r
a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W &Ccedil;&aacute;&icirc; c I ∇ ⋅ c
r
r
`&igrave;&ecirc;&auml; &ccedil;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W &Aring;&igrave;&ecirc;&auml; c I ∇ &times; c
i~&eacute;&auml;~&Aring;&aacute;~&aring; &ccedil;&eacute;&Eacute;&ecirc;~&iacute;&ccedil;&ecirc;W ∇ O
856. d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute; &ccedil;&Ntilde; ~ p&Aring;~&auml;~&ecirc; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
 ∂&Ntilde; ∂&Ntilde; ∂&Ntilde; 
&Ouml;&ecirc;~&Ccedil; &Ntilde; = ∇&Ntilde; =  I I  I
 ∂&ntilde; ∂&oacute; ∂&ograve; 
 ∂&igrave; ∂&igrave;
∂&igrave; 
K
&Ouml;&ecirc;~&Ccedil; &igrave; = ∇&igrave; = 
I
IKI
∂&ntilde; &aring; 
 ∂&ntilde; N ∂&ntilde; O
857. a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;~&auml; a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;
∂&Ntilde;
∂&Ntilde;
∂&Ntilde; ∂&Ntilde;
= &Aring;&ccedil;&euml; α + &Aring;&ccedil;&euml; β + &Aring;&ccedil;&euml; γ I
∂&oacute;
∂&ograve;
∂&auml; ∂&ntilde;
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&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;
r
&auml; (&Aring;&ccedil;&euml; αI &Aring;&ccedil;&euml; βI &Aring;&ccedil;&euml; γ ) I &Aring;&ccedil;&euml; O α + &Aring;&ccedil;&euml; O β + &Aring;&ccedil;&euml; O γ = N K
858. a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; ~ s&Eacute;&Aring;&iacute;&ccedil;&ecirc; c&aacute;&Eacute;&auml;&Ccedil;
r
r ∂m ∂n ∂o
+
+
&Ccedil;&aacute;&icirc; c = ∇ ⋅ c =
∂&ntilde; ∂&oacute; ∂&ograve;
859. `&igrave;&ecirc;&auml; &ccedil;&Ntilde; ~ s&Eacute;&Aring;&iacute;&ccedil;&ecirc; c&aacute;&Eacute;&auml;&Ccedil;
r
r
&aacute;
&agrave;
r
r ∂
∂
&Aring;&igrave;&ecirc;&auml; c = ∇ &times; c =
∂&ntilde; ∂&ntilde;
m n
r
&acirc;
∂
∂&ntilde;
o
 ∂o ∂n  r  ∂m ∂o  r  ∂n ∂m  r
 &aacute; +  −
= 
−
− &acirc;
 &agrave; + 
 ∂&oacute; ∂&ograve;   ∂&ograve; ∂&ntilde;   ∂&ntilde; ∂&oacute; 
860. i~&eacute;&auml;~&Aring;&aacute;~&aring; l&eacute;&Eacute;&ecirc;~&iacute;&ccedil;&ecirc;
∂ O&Ntilde; ∂ O&Ntilde; ∂ O&Ntilde;
∇ O&Ntilde; = O + O + O
∂&ograve;
∂&oacute;
∂&ntilde;
(
)
(
)
r
r
861. &Ccedil;&aacute;&icirc; &Aring;&igrave;&ecirc;&auml; c = ∇ ⋅ ∇ &times; c ≡ M
862. &Aring;&igrave;&ecirc;&auml;(&Ouml;&ecirc;~&Ccedil; &Ntilde; ) = ∇ &times; (∇&Ntilde; ) ≡ M
863. &Ccedil;&aacute;&icirc; (&Ouml;&ecirc;~&Ccedil; &Ntilde; ) = ∇ ⋅ (∇&Ntilde; ) = ∇ O&Ntilde;
(
)
(
)
( )
r
r
r
r
r
864. &Aring;&igrave;&ecirc;&auml; &Aring;&igrave;&ecirc;&auml; c = &Ouml;&ecirc;~&Ccedil; &Ccedil;&aacute;&icirc; c − ∇ Oc = ∇ ∇ ⋅ c − ∇ Oc
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Chapter 9
Integral Calculus
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde;I &Ouml;I &igrave;I &icirc;
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &ntilde;I &iacute;I ξ
f&aring;&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; I ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde; I &pound;
a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; &ccedil;&Ntilde; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W &oacute; ′(&ntilde; ) I &Ntilde; ′(&ntilde; ) I c′(&ntilde; ) I &pound;
o&Eacute;~&auml; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;W `I ~I &Auml;I &Aring;I &Ccedil;I &acirc;
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &atilde;I &aring;I &aacute;I &agrave;
9.1 Indefinite Integral
865.
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = c(&ntilde; ) + ` &aacute;&Ntilde; c′(&ntilde; ) = &Ntilde; (&ntilde; ) K
866.
(∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; ) ′= &Ntilde; (&ntilde; )
867.
∫ &acirc;&Ntilde; (&ntilde; )&Ccedil;&ntilde; = &acirc; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde;
868.
∫ [&Ntilde; (&ntilde; ) + &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde;
869.
∫ [&Ntilde; (&ntilde; ) − &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; − ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde;
870.
∫ &Ntilde; (~&ntilde; )&Ccedil;&ntilde; = ~ c(~&ntilde; ) + `
N
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CHAPTER 9. INTEGRAL CALCULUS
N
871.
∫ &Ntilde; (~&ntilde; + &Auml;)&Ccedil;&ntilde; = ~ c(~&ntilde; + &Auml;) + `
872.
N
∫ &Ntilde; (&ntilde; )&Ntilde; ′(&ntilde; )&Ccedil;&ntilde; = O &Ntilde; (&ntilde; ) + `
873.
∫ &Ntilde; (&ntilde; ) &Ccedil;&ntilde; = &auml;&aring; &Ntilde; (&ntilde; ) + `
O
&Ntilde; ′(&ntilde; )
874. j&Eacute;&iacute;&Uuml;&ccedil;&Ccedil; &ccedil;&Ntilde; p&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&igrave;(&iacute; ))&igrave;′(&iacute; )&Ccedil;&iacute; &aacute;&Ntilde; &ntilde; = &igrave;(&iacute; ) K
875. f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring; &Auml;&oacute; m~&ecirc;&iacute;&euml;
∫ &igrave;&Ccedil;&icirc; = &igrave;&icirc; − ∫ &icirc;&Ccedil;&igrave; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &igrave;(&ntilde; ) I &icirc; (&ntilde; ) ~&ecirc;&Eacute; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;K
9.2 Integrals of Rational Functions
876.
∫ ~&Ccedil;&ntilde; = ~&ntilde; + `
877.
∫ &ntilde;&Ccedil;&ntilde; =
878.
O
∫ &ntilde; &Ccedil;&ntilde; =
&ntilde;P
+`
P
879.
&eacute;
∫ &ntilde; &Ccedil;&ntilde; =
&ntilde; &eacute; +N
+ ` I &eacute; ≠ −N K
&eacute;+N
&ntilde;O
+`
O
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CHAPTER 9. INTEGRAL CALCULUS
880.
&aring; +N
(
~&ntilde; + &Auml;)
∫ (~&ntilde; + &Auml;) &Ccedil;&ntilde; = ~(&aring; + N) + ` I &aring; ≠ −N K
881.
∫
882.
∫ ~&ntilde; + &Auml; = ~ &auml;&aring; ~&ntilde; + &Auml; + `
883.
∫ &Aring;&ntilde; + &Ccedil; &Ccedil;&ntilde; = &Aring; &ntilde; +
884.
∫ (&ntilde; + ~ )(&ntilde; + &Auml;) = ~ − &Auml; &auml;&aring; &ntilde; + ~ + ` I ~ ≠ &Auml; K
885.
N
&ntilde;&Ccedil;&ntilde;
∫ ~ + &Auml;&ntilde; = &Auml; (~ + &Auml;&ntilde; − ~ &auml;&aring; ~ + &Auml;&ntilde; ) + `
886.
&ntilde; O&Ccedil;&ntilde;
N N

O
O
∫ ~ + &Auml;&ntilde; = &Auml;P  O (~ + &Auml;&ntilde; ) − O~(~ + &Auml;&ntilde; ) + ~ &auml;&aring; ~ + &Auml;&ntilde;  + `
887.
∫ &ntilde; (~ + &Auml;&ntilde; ) = ~ &auml;&aring;
888.
∫ &ntilde; (~ + &Auml;&ntilde; ) = − ~&ntilde; + ~
889.
∫ (~ + &Auml;&ntilde; )
&aring;
&Ccedil;&ntilde;
= &auml;&aring; &ntilde; + `
&ntilde;
&Ccedil;&ntilde;
N
~&ntilde; + &Auml;
~
&Ccedil;&ntilde;
&Auml;&Aring; − ~&Ccedil;
&auml;&aring; &Aring;&ntilde; + &Ccedil; + `
&Aring;O
&ntilde;+&Auml;
N
O
&Ccedil;&ntilde;
N
&Ccedil;&ntilde;
~ + &Auml;&ntilde;
+`
&ntilde;
N
O
&ntilde;&Ccedil;&ntilde;
O
=
&Auml;
O
&auml;&aring;
~ + &Auml;&ntilde;
+`
&ntilde;
N
~ 
&auml;&aring; ~ + &Auml;&ntilde; +
+`
O 
&Auml; 
~ + &Auml;&ntilde; 
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CHAPTER 9. INTEGRAL CALCULUS
890.
~O 
&ntilde; O&Ccedil;&ntilde;
N

+`
=
+
−
+
−
~
&Auml;&ntilde;
O
~
&auml;&aring;
~
&Auml;&ntilde;
∫ (~ + &Auml;&ntilde; )O &Auml;P 
~ + &Auml;&ntilde; 
891.
∫ &ntilde; (~ + &Auml;&ntilde; )
892.
∫&ntilde;
893.
∫ N− &ntilde;
894.
∫~
O
895.
∫&ntilde;
O
896.
∫ N+ &ntilde;
897.
∫~
O
898.
∫&ntilde;
O
899.
∫ ~ + &Auml;&ntilde;
&Ccedil;&ntilde;
O
=
N
N ~ + &Auml;&ntilde;
+ O &auml;&aring;
+`
~ (~ + &Auml;&ntilde; ) ~
&ntilde;
&Ccedil;&ntilde;
N &ntilde; −N
+`
= &auml;&aring;
−N O &ntilde; +N
O
&Ccedil;&ntilde;
O
N N+ &ntilde;
= &auml;&aring;
+`
O N− &ntilde;
&Ccedil;&ntilde;
N ~+&ntilde;
+`
= &auml;&aring;
O
O~ ~ − &ntilde;
−&ntilde;
&ntilde;−~
&Ccedil;&ntilde;
N
+`
= &auml;&aring;
O
O~ &ntilde; + ~
−~
&Ccedil;&ntilde;
O
= &iacute;~&aring; −N &ntilde; + `
&Ccedil;&ntilde;
&ntilde;
N
= &iacute;~&aring; −N + `
O
~
~
+&ntilde;
&ntilde;&Ccedil;&ntilde;
N
= &auml;&aring;(&ntilde; O + ~ O ) + `
O
+~
O
&Ccedil;&ntilde;
O
=
 &Auml;
N
 + ` I ~&Auml; &gt; M K
~&ecirc;&Aring;&iacute;~&aring; &ntilde;

~
~&Auml;


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CHAPTER 9. INTEGRAL CALCULUS
&ntilde;&Ccedil;&ntilde;
~
N
&auml;&aring; &ntilde; O + + `
&Auml;
O&Auml;
900.
∫ ~ + &Auml;&ntilde;
901.
&ntilde;O
&Ccedil;&ntilde;
N
=
&auml;&aring;
∫ &ntilde; (~ + &Auml;&ntilde; O ) O~ ~ + &Auml;&ntilde; O + `
902.
∫~
O
O
=
&Ccedil;&ntilde;
N
~ + &Auml;&ntilde;
=
&auml;&aring;
+`
O O
−&Auml; &ntilde;
O~&Auml; ~ − &Auml;&ntilde;
O~&ntilde; + &Auml; − &Auml;O − Q~&Aring;
&Ccedil;&ntilde;
N
903. ∫ O
=
+`I
&auml;&aring;
~&ntilde; + &Auml;&ntilde; + &Aring;
&Auml;O − Q~&Aring; O~&ntilde; + &Auml; + &Auml;O − Q~&Aring;
&Auml;O − Q~&Aring; &gt; M K
904.
O~&ntilde; + &Auml;
&Ccedil;&ntilde;
O
=
+` I
~&ecirc;&Aring;&iacute;~&aring;
+ &Auml;&ntilde; + &Aring;
Q~&Aring; − &Auml;O
Q~&Aring; − &Auml;O
&Auml;O − Q~&Aring; &lt; M K
∫ ~&ntilde;
O
9.3 Integrals of Irrational Functions
905.
∫
&Ccedil;&ntilde;
O
~&ntilde; + &Auml; + `
=
~&ntilde; + &Auml; ~
906.
∫
~&ntilde; + &Auml; &Ccedil;&ntilde; =
907.
∫
&ntilde;&Ccedil;&ntilde;
O(~&ntilde; − O&Auml;)
=
~&ntilde; + &Auml; + `
P~ O
~&ntilde; + &Auml;
P
O
(~&ntilde; + &Auml;) O + `
P~
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CHAPTER 9. INTEGRAL CALCULUS
908.
909.
910.
∫&ntilde;
~&ntilde; + &Auml; &Ccedil;&ntilde; =
P
O(P~&ntilde; − O&Auml;)
(~&ntilde; + &Auml;) O + `
O
NR~
~&ntilde; + &Auml; − &Auml; − ~&Aring;
&Ccedil;&ntilde;
N
&auml;&aring;
=
+`I
~&ntilde; + &Auml; + &Auml; − ~&Aring;
~&ntilde; + &Auml;
&Auml; − ~&Aring;
&Auml; − ~&Aring; &gt; M K
∫ (&ntilde; + &Aring; )
~&ntilde; + &Auml;
&Ccedil;&ntilde;
N
~&ecirc;&Aring;&iacute;~&aring;
=
+`I
~&Aring; − &Auml;
~&ntilde; + &Auml;
~&Aring; − &Auml;
&Auml; − ~&Aring; &lt; M K
∫ (&ntilde; + &Aring; )
911.
∫
~&ntilde; + &Auml;
N
(~&ntilde; + &Auml;)(&Aring;&ntilde; + &Ccedil; ) −
&Ccedil;&ntilde; =
&Aring;&ntilde; + &Ccedil;
&Aring;
~&Ccedil; − &Auml;&Aring;
&auml;&aring; ~ (&Aring;&ntilde; + &Ccedil; ) + &Aring;(~&ntilde; + &Auml;) + ` I ~ &gt; M K
−
&Aring; ~&Aring;
912.
∫
~&ntilde; + &Auml;
N
&Ccedil;&ntilde; =
&Aring;&ntilde; + &Ccedil;
&Aring;
−
913.
~ (&Aring;&ntilde; + &Ccedil; )
~&Ccedil; − &Auml;&Aring;
~&ecirc;&Aring;&iacute;~&aring;
+ ` I E ~ &lt; M I &Aring; &gt; M FK
&Aring;(~&ntilde; + &Auml;)
&Aring; ~&Aring;
O
∫ &ntilde; ~ + &Auml;&ntilde; &Ccedil;&ntilde; =
914.
∫
915.
∫&ntilde;
(~&ntilde; + &Auml;)(&Aring;&ntilde; + &Ccedil; ) −
O(U~ O − NO~&Auml;&ntilde; + NR&Auml;O &ntilde; O )
NMR&Auml;P
(~ + &Auml;&ntilde; )P + `
&ntilde; O&Ccedil;&ntilde;
O(U~ O − Q~&Auml;&ntilde; + P&Auml;O &ntilde; O )
=
~ + &Auml;&ntilde; + `
NR&Auml;P
~ + &Auml;&ntilde;
&Ccedil;&ntilde;
N
~ + &Auml;&ntilde; − ~
=
&auml;&aring;
+`I ~ &gt;MK
~ + &Auml;&ntilde;
~
~ + &Auml;&ntilde; + ~
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CHAPTER 9. INTEGRAL CALCULUS
&Ccedil;&ntilde;
O
~ + &Auml;&ntilde;
=
~&ecirc;&Aring;&iacute;~&aring;
+`I ~ &lt;MK
−~
~ + &Auml;&ntilde;
−~
916.
∫&ntilde;
917.
∫
~−&ntilde;
&Ccedil;&ntilde; =
&Auml;+ &ntilde;
918.
∫
~+&ntilde;
&Auml;−&ntilde;
&Ccedil;&ntilde; = − (~ + &ntilde; )(&Auml; − &ntilde; ) − (~ + &Auml;)~&ecirc;&Aring;&euml;&aacute;&aring;
+`
&Auml;−&ntilde;
~+&Auml;
919.
∫
N+ &ntilde;
&Ccedil;&ntilde; = − N − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; + `
N− &ntilde;
920.
∫ (&ntilde; − ~ )(&Auml; − ~ ) = O ~&ecirc;&Aring;&euml;&aacute;&aring;
921.
∫
922.
∫
(~ − &ntilde; )(&Auml; + &ntilde; ) + (~ + &Auml;)~&ecirc;&Aring;&euml;&aacute;&aring;
&Ccedil;&ntilde;
∫
924.
∫
&ntilde; −~
+`
&Auml;−~
O&Aring;&ntilde; − &Auml;
~ + &Auml;&ntilde; − &Aring;&ntilde; O +
Q&Aring;
&Auml;O − Q~&Aring;
O&Aring;&ntilde; − &Auml;
+
~&ecirc;&Aring;&euml;&aacute;&aring;
+`
U &Aring;P
&Auml;O + Q~&Aring;
~ + &Auml;&ntilde; − &Aring;&ntilde; O &Ccedil;&ntilde; =
&Ccedil;&ntilde;
~&ntilde; O + &Auml;&ntilde; + &Aring;
~ &gt; MK
923.
&ntilde;+&Auml;
+`
~+&Auml;
&Ccedil;&ntilde;
~&ntilde; + &Auml;&ntilde; + &Aring;
O
&ntilde; O + ~ O &Ccedil;&ntilde; =
=
N
&auml;&aring; O~&ntilde; + &Auml; + O ~ (~&ntilde; O + &Auml;&ntilde; + &Aring; ) + ` I
~
=−
N
O~&ntilde; + &Auml; O
~&ecirc;&Aring;&euml;&aacute;&aring;
&Auml; − Q~&Aring; + ` I ~ &lt; M K
Q~
~
&ntilde; O O ~O
&ntilde; + ~ + &auml;&aring; &ntilde; + &ntilde; O + ~ O + `
O
O
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CHAPTER 9. INTEGRAL CALCULUS
925.
∫&ntilde;
926.
∫&ntilde;
&ntilde; O + ~ O &Ccedil;&ntilde; =
&ntilde; O + ~ O &Ccedil;&ntilde; =
O
−
927.
∫
928.
∫
929.
∫
930.
∫
931.
∫
932.
∫&ntilde;
933.
∫
934.
∫&ntilde;
N O O PO
(&ntilde; + ~ ) + `
P
&ntilde;
(
O&ntilde; O + ~ O ) &ntilde; O + ~ O −
U
~Q
&auml;&aring; &ntilde; + &ntilde; O + ~ O + `
U
&ntilde; O + ~O
&ntilde; O + ~O
&Ccedil;&ntilde;
=
−
+ &auml;&aring; &ntilde; + &ntilde; O + ~ O + `
&ntilde;O
&ntilde;
&Ccedil;&ntilde;
&ntilde; +~
O
O
= &auml;&aring; &ntilde; + &ntilde; O + ~ O + `
&ntilde; O + ~O
&ntilde;
&Ccedil;&ntilde; = &ntilde; O + ~ O + ~ &auml;&aring;
+`
&ntilde;
~ + &ntilde; O + ~O
&ntilde;&Ccedil;&ntilde;
&ntilde; +~
O
O
&ntilde; O&Ccedil;&ntilde;
&ntilde; O + ~O
= &ntilde; O + ~O + `
=
&ntilde; O O ~O
&ntilde; + ~ − &auml;&aring; &ntilde; + &ntilde; O + ~ O + `
O
O
&Ccedil;&ntilde;
N
&ntilde;
= &auml;&aring;
+`
&ntilde; O + ~O ~ ~ + &ntilde; O + ~O
&ntilde; O − ~ O &Ccedil;&ntilde; =
&ntilde; O O ~O
&ntilde; − ~ − &auml;&aring; &ntilde; + &ntilde; O − ~ O + `
O
O
&ntilde; O − ~ O &Ccedil;&ntilde; =
N O O PO
(&ntilde; − ~ ) + `
P
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CHAPTER 9. INTEGRAL CALCULUS
∫
&ntilde;O − ~O
~
&Ccedil;&ntilde; = &ntilde; O − ~ O + ~ ~&ecirc;&Aring;&euml;&aacute;&aring; + `
&ntilde;
&ntilde;
936.
∫
&ntilde;O − ~O
&ntilde; O − ~O
&Ccedil;&ntilde;
=−
+ &auml;&aring; &ntilde; + &ntilde; O − ~ O + `
O
&ntilde;
&ntilde;
937.
∫
938.
∫
935.
&Ccedil;&ntilde;
&ntilde; −~
O
= &auml;&aring; &ntilde; + &ntilde; O − ~ O + `
O
&ntilde;&Ccedil;&ntilde;
&ntilde; −~
O
= &ntilde;O − ~O + `
O
&ntilde; O&Ccedil;&ntilde;
&ntilde; O O ~O
=
&ntilde; − ~ + &auml;&aring; &ntilde; + &ntilde; O − ~ O + `
O
O
O
O
&ntilde; −~
939.
∫
940.
∫&ntilde;
941.
∫ (&ntilde; + ~ )
942.
∫ (&ntilde; − ~ )
&Ccedil;&ntilde;
N
~
= − ~&ecirc;&Aring;&euml;&aacute;&aring; + `
~
&ntilde;
&ntilde; −~
O
O
&Ccedil;&ntilde;
&ntilde; O − ~O
&Ccedil;&ntilde;
&ntilde; −~
O
&Ccedil;&ntilde;
943.
∫&ntilde;
944.
∫ (&ntilde;
O
&ntilde; O − ~O
&Ccedil;&ntilde;
O
− ~O ) O
P
O
=
=−
=
N &ntilde; −~
+`
~ &ntilde;+~
=−
N &ntilde;+~
+`
~ &ntilde; −~
&ntilde; O − ~O
+`
~O&ntilde;
&ntilde;
~O &ntilde; O − ~O
+`
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CHAPTER 9. INTEGRAL CALCULUS
945.
∫ (&ntilde;
O
− ~ O ) O &Ccedil;&ntilde; = −
P
&ntilde;
(
O&ntilde; O − R~ O ) &ntilde; O − ~ O +
U
P~ Q
+
&auml;&aring; &ntilde; + &ntilde; O − ~ O + `
U
&ntilde; O
~O
&ntilde;
~ − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; + `
O
~
O
946.
∫
947.
O
O
∫ &ntilde; ~ − &ntilde; &Ccedil;&ntilde; = −
948.
O
O
O
∫ &ntilde; ~ − &ntilde; &Ccedil;&ntilde; =
~ O − &ntilde; O &Ccedil;&ntilde; =
P
N O
(
~ − &ntilde;O ) O + `
P
&ntilde;
~Q
&ntilde;
(
O&ntilde; O − ~ O ) ~ O − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; + `
U
U
~
∫
~O − &ntilde; O
&ntilde;
&Ccedil;&ntilde; = ~ O − &ntilde; O + ~ &auml;&aring;
+`
&ntilde;
~ + ~O − &ntilde; O
950.
∫
~O − &ntilde;O
~O − &ntilde;O
&ntilde;
&Ccedil;&ntilde;
=
−
− ~&ecirc;&Aring;&euml;&aacute;&aring; + `
O
&ntilde;
&ntilde;
~
951.
∫
952.
∫
953.
∫
949.
954.
∫
&Ccedil;&ntilde;
= ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; + `
N− &ntilde;O
&Ccedil;&ntilde;
~O − &ntilde;O
&ntilde;&Ccedil;&ntilde;
~ −&ntilde;
O
O
&ntilde; O&Ccedil;&ntilde;
~O − &ntilde; O
= &euml;&aacute;&aring;
&ntilde;
+`
~
= − ~O − &ntilde; O + `
=−
&ntilde; O
~O
&ntilde;
~ − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; + `
O
O
~
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&Ccedil;&ntilde;
955.
∫ (&ntilde; + ~ )
956.
∫ (&ntilde; − ~ )
~O − &ntilde; O
&Ccedil;&ntilde;
~O − &ntilde; O
&Ccedil;&ntilde;
957.
∫ (&ntilde; + &Auml; )
958.
∫ (&ntilde; + &Auml; )
~O − &ntilde; O
&Ccedil;&ntilde;
~ −&ntilde;
O
O
=−
N ~−&ntilde;
+`
O ~+&ntilde;
=−
N ~+&ntilde;
+`
O ~−&ntilde;
N
=
&Auml;O − ~ O
N
=
~ −&Auml;
O
O
~&ecirc;&Aring;&euml;&aacute;&aring;
&auml;&aring;
&Auml;&ntilde; + ~ O
+`I &Auml;&gt;~ K
~ (&ntilde; + &Auml; )
&ntilde;+&Auml;
~ −&Auml;
O
O
~ O − &ntilde; O + ~ O + &Auml;&ntilde;
&Auml;&lt;~ K
&Ccedil;&ntilde;
959.
∫&ntilde;
960.
∫ (~
961.
∫ (~
O
O
~O − &ntilde;O
− &ntilde; O ) O &Ccedil;&ntilde; =
P
&Ccedil;&ntilde;
O
=−
−&ntilde;
O
)
P
=
O
~O − &ntilde;O
+`
~O&ntilde;
&ntilde; O
P~ Q
&ntilde;
(
R~ − O&ntilde; O ) ~ O − &ntilde; O +
~&ecirc;&Aring;&euml;&aacute;&aring; + `
U
U
~
&ntilde;
~O ~O − &ntilde; O
+`
9.4 Integrals of Trigonometric Functions
962.
∫ &euml;&aacute;&aring; &ntilde;&Ccedil;&ntilde; = − &Aring;&ccedil;&euml; &ntilde; + `
963.
∫ &Aring;&ccedil;&euml; &ntilde;&Ccedil;&ntilde; = &euml;&aacute;&aring; &ntilde; + `
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CHAPTER 9. INTEGRAL CALCULUS
&ntilde; &Ccedil;&ntilde; =
&ntilde; N
− &euml;&aacute;&aring; O&ntilde; + `
O Q
&ntilde; &Ccedil;&ntilde; =
&ntilde; N
+ &euml;&aacute;&aring; O&ntilde; + `
O Q
964.
∫ &euml;&aacute;&aring;
965.
∫ &Aring;&ccedil;&euml;
966.
∫ &euml;&aacute;&aring;
967.
∫ &Aring;&ccedil;&euml;
968.
∫ &euml;&aacute;&aring; &ntilde; = ∫ &Aring;&euml;&Aring; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; O + `
969.
∫ &Aring;&ccedil;&euml; &ntilde; = ∫ &euml;&Eacute;&Aring; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; O + Q  + `
970.
∫ &euml;&aacute;&aring;
971.
∫ &Aring;&ccedil;&euml;
972.
∫ &euml;&aacute;&aring;
973.
∫ &Aring;&ccedil;&euml;
974.
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = − Q &Aring;&ccedil;&euml; O&ntilde; + `
O
O
P
N
N
P
&ntilde; &Ccedil;&ntilde; = &Aring;&ccedil;&euml;P &ntilde; − &Aring;&ccedil;&euml; &ntilde; + ` = &Aring;&ccedil;&euml; P&ntilde; − &Aring;&ccedil;&euml; &ntilde; + `
NO
Q
P
P
N
N
P
&ntilde; &Ccedil;&ntilde; = &euml;&aacute;&aring; &ntilde; − &euml;&aacute;&aring; P &ntilde; + ` = &euml;&aacute;&aring; P&ntilde; + &euml;&aacute;&aring; &ntilde; + `
P
NO
Q
&Ccedil;&ntilde;
&ntilde;
&Ccedil;&ntilde;
&ntilde;
&Ccedil;&ntilde;
O
&ntilde;
= ∫ &Aring;&euml;&Aring; O &ntilde; &Ccedil;&ntilde; = − &Aring;&ccedil;&iacute; &ntilde; + `
&ntilde;
= ∫ &euml;&Eacute;&Aring; O &ntilde; &Ccedil;&ntilde; = &iacute;~&aring; &ntilde; + `
&ntilde;
= ∫ &Aring;&euml;&Aring; P &ntilde; &Ccedil;&ntilde; = −
&ntilde;
= ∫ &euml;&Eacute;&Aring; P &ntilde; &Ccedil;&ntilde; =
&Ccedil;&ntilde;
O
&Ccedil;&ntilde;
P
&Ccedil;&ntilde;
P
π
&Aring;&ccedil;&euml; &ntilde;
N
&ntilde;
+ &auml;&aring; &iacute;~&aring; + `
O
O &euml;&aacute;&aring; &ntilde; O
O
&euml;&aacute;&aring; &ntilde;
N
 &ntilde; π
+ &auml;&aring; &iacute;~&aring; +
+`
O
O &Aring;&ccedil;&euml; &ntilde; O
O Q
N
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CHAPTER 9. INTEGRAL CALCULUS
N
&ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = &euml;&aacute;&aring; P &ntilde; + `
P
975.
∫ &euml;&aacute;&aring;
976.
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml;
977.
∫ &euml;&aacute;&aring;
978.
∫ &iacute;~&aring; &ntilde;&Ccedil;&ntilde; = − &auml;&aring; &Aring;&ccedil;&euml; &ntilde; + `
979.
∫ &Aring;&ccedil;&euml;
980.
&euml;&aacute;&aring; O &ntilde;
 &ntilde; π
∫ &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; O + Q  − &euml;&aacute;&aring; &ntilde; + `
981.
∫ &iacute;~&aring;
982.
∫ &Aring;&ccedil;&iacute; &ntilde;&Ccedil;&ntilde; = &auml;&aring; &euml;&aacute;&aring; &ntilde; + `
983.
∫ &euml;&aacute;&aring;
O
O
O
N
&ntilde; &Ccedil;&ntilde; = − &Aring;&ccedil;&euml;P &ntilde; + `
P
&ntilde; &Aring;&ccedil;&euml; O &ntilde; &Ccedil;&ntilde; =
&ntilde; N
− &euml;&aacute;&aring; Q &ntilde; + `
U PO
&euml;&aacute;&aring; &ntilde;
N
+ ` = &euml;&Eacute;&Aring; &ntilde; + `
&Ccedil;&ntilde; =
O
&ntilde;
&Aring;&ccedil;&euml; &ntilde;
O
&ntilde; &Ccedil;&ntilde; = &iacute;~&aring; &ntilde; − &ntilde; + `
&Aring;&ccedil;&euml; &ntilde;
N
+ ` = − &Aring;&euml;&Aring; &ntilde; + `
&Ccedil;&ntilde; = −
O
&ntilde;
&euml;&aacute;&aring; &ntilde;
&Aring;&ccedil;&euml; O &ntilde;
&ntilde;
&Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; + &Aring;&ccedil;&euml; &ntilde; + `
984. ∫
&euml;&aacute;&aring; &ntilde;
O
985.
∫ &Aring;&ccedil;&iacute;
986.
∫ &Aring;&ccedil;&euml; &ntilde; &euml;&aacute;&aring; &ntilde; = &auml;&aring; &iacute;~&aring; &ntilde; + `
O
&ntilde; &Ccedil;&ntilde; = − &Aring;&ccedil;&iacute; &ntilde; − &ntilde; + `
&Ccedil;&ntilde;
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CHAPTER 9. INTEGRAL CALCULUS
&Ccedil;&ntilde;
N
 &ntilde; π
=−
+ &auml;&aring; &iacute;~&aring; +  + `
&ntilde; &Aring;&ccedil;&euml; &ntilde;
&euml;&aacute;&aring; &ntilde;
O Q
987.
∫ &euml;&aacute;&aring;
988.
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml;
989.
∫ &euml;&aacute;&aring;
990.
∫ &euml;&aacute;&aring; &atilde;&ntilde; &euml;&aacute;&aring; &aring;&ntilde; &Ccedil;&ntilde; = − O(&atilde; + &aring;)
O
&Ccedil;&ntilde;
O
O
&ntilde;
=
N
&ntilde;
+ &auml;&aring; &iacute;~&aring; + `
&Aring;&ccedil;&euml; &ntilde;
O
&Ccedil;&ntilde;
= &iacute;~&aring; &ntilde; − &Aring;&ccedil;&iacute; &ntilde; + `
&ntilde; &Aring;&ccedil;&euml; O &ntilde;
&euml;&aacute;&aring;(&atilde; + &aring; )&ntilde;
+
&euml;&aacute;&aring;(&atilde; − &aring; )&ntilde;
+`I
O(&atilde; − &aring; )
−
&Aring;&ccedil;&euml;(&atilde; − &aring; )&ntilde;
+`I
O(&atilde; − &aring; )
&atilde;O ≠ &aring;O K
991.
&Aring;&ccedil;&euml;(&atilde; + &aring; )&ntilde;
∫ &euml;&aacute;&aring; &atilde;&ntilde; &Aring;&ccedil;&euml; &aring;&ntilde; &Ccedil;&ntilde; = − O(&atilde; + &aring;)
&atilde;O ≠ &aring;O K
992.
&euml;&aacute;&aring;(&atilde; + &aring; )&ntilde;
∫ &Aring;&ccedil;&euml; &atilde;&ntilde; &Aring;&ccedil;&euml; &aring;&ntilde; &Ccedil;&ntilde; = O(&atilde; + &aring;)
+
&euml;&aacute;&aring;(&atilde; − &aring; )&ntilde;
+`I
O(&atilde; − &aring; )
&atilde;O ≠ &aring;O K
993.
∫ &euml;&Eacute;&Aring; &ntilde; &iacute;~&aring; &ntilde;&Ccedil;&ntilde; = &euml;&Eacute;&Aring; &ntilde; + `
994.
∫ &Aring;&euml;&Aring; &ntilde; &Aring;&ccedil;&iacute; &ntilde;&Ccedil;&ntilde; = − &Aring;&euml;&Aring; &ntilde; + `
995.
&aring;
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = −
&Aring;&ccedil;&euml; &aring;+N &ntilde;
+`
&aring; +N
&euml;&aacute;&aring; &aring; +N &ntilde;
996. ∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; =
+`
&aring; +N
&aring;
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997.
∫ ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; +
N− &ntilde;O + `
998.
∫ ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; −
N− &ntilde;O + `
999.
∫ ~&ecirc;&Aring;&iacute;~&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring;&iacute;~&aring; &ntilde; − O &auml;&aring;(&ntilde;
N
O
+ N) + `
N
1000. ∫ ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; + &auml;&aring;(&ntilde; O + N) + `
O
9.5 Integrals of Hyperbolic Functions
1001. ∫ &euml;&aacute;&aring;&Uuml; &ntilde;&Ccedil;&ntilde; = &Aring;&ccedil;&euml;&Uuml; &ntilde; + `
1002. ∫ &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde; = &euml;&aacute;&aring;&Uuml; &ntilde; + `
1003. ∫ &iacute;~&aring;&Uuml; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &Aring;&ccedil;&euml;&Uuml; &ntilde; + `
1004. ∫ &Aring;&ccedil;&iacute;&Uuml; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &euml;&aacute;&aring;&Uuml; &ntilde; + `
1005. ∫ &euml;&Eacute;&Aring;&Uuml; O &ntilde;&Ccedil;&ntilde; = &iacute;~&aring;&Uuml; &ntilde; + `
1006. ∫ &Aring;&euml;&Aring;&Uuml; O &ntilde;&Ccedil;&ntilde; = − &Aring;&ccedil;&iacute;&Uuml; &ntilde; + `
1007. ∫ &euml;&Eacute;&Aring;&Uuml;&ntilde; &iacute;~&aring;&Uuml; &ntilde;&Ccedil;&ntilde; = −&euml;&Eacute;&Aring;&Uuml;&ntilde; + `
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1008. ∫ &Aring;&euml;&Aring;&Uuml;&ntilde; &Aring;&ccedil;&iacute;&Uuml; &ntilde;&Ccedil;&ntilde; = −&Aring;&euml;&Aring;&Uuml;&ntilde; + `
9.6 Integrals of Exponential and Logarithmic
Functions
1009. ∫ &Eacute; &ntilde; &Ccedil;&ntilde; = &Eacute; &ntilde; + `
~&ntilde;
1010. ∫ ~ &Ccedil;&ntilde; =
+`
&auml;&aring; ~
&ntilde;
1011. ∫ &Eacute; ~&ntilde; &Ccedil;&ntilde; =
&Eacute; ~&ntilde;
+`
~
1012. ∫ &ntilde;&Eacute; ~&ntilde; &Ccedil;&ntilde; =
&Eacute; ~&ntilde;
(~&ntilde; − N) + `
~O
1013. ∫ &auml;&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; &auml;&aring; &ntilde; − &ntilde; + `
1014.
&Ccedil;&ntilde;
∫ &ntilde; &auml;&aring; &ntilde; = &auml;&aring; &auml;&aring; &ntilde; + `
 &auml;&aring; &ntilde;
N 
1015. ∫ &ntilde; &aring; &auml;&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; &aring; +N 
−
+`
O
 &aring; + N (&aring; + N) 
1016. ∫ &Eacute; ~&ntilde; &euml;&aacute;&aring; &Auml;&ntilde; &Ccedil;&ntilde; =
~ &euml;&aacute;&aring; &Auml;&ntilde; − &Auml; &Aring;&ccedil;&euml; &Auml;&ntilde; ~&ntilde;
&Eacute; +`
~ O + &Auml;O
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CHAPTER 9. INTEGRAL CALCULUS
1017. ∫ &Eacute; ~&ntilde; &Aring;&ccedil;&euml; &Auml;&ntilde; &Ccedil;&ntilde; =
~ &Aring;&ccedil;&euml; &Auml;&ntilde; + &Auml; &euml;&aacute;&aring; &Auml;&ntilde; ~&ntilde;
&Eacute; +`
~ O + &Auml;O
9.7 Reduction Formulas
1018. ∫ &ntilde; &aring; &Eacute; &atilde;&ntilde; &Ccedil;&ntilde; =
1019.
N &aring; &atilde;&ntilde; &aring;
&ntilde; &Eacute; − ∫ &ntilde; &aring; −N&Eacute; &atilde;&ntilde; &Ccedil;&ntilde;
&atilde;
&atilde;
&Eacute; &atilde;&ntilde;
&Eacute; &atilde;&ntilde;
&atilde; &Eacute; &atilde;&ntilde;
&Ccedil;&ntilde;
=
−
+
&Ccedil;&ntilde; I &aring; ≠ N K
∫ &ntilde;&aring;
(&aring; − N)&ntilde; &aring;−N &aring; − N ∫ &ntilde; &aring;−N
1020. ∫ &euml;&aacute;&aring;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; =
1021.
&Ccedil;&ntilde;
∫ &euml;&aacute;&aring;&Uuml;
&aring;
&ntilde;
=−
1022. ∫ &Aring;&ccedil;&euml;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; =
1023.
&Ccedil;&ntilde;
∫ &Aring;&ccedil;&euml;&Uuml;
&aring;
&ntilde;
=−
N
&aring; −N
&euml;&aacute;&aring;&Uuml; &aring; −N &ntilde; &Aring;&ccedil;&euml;&Uuml; &ntilde; −
&euml;&aacute;&aring;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde;
∫
&aring;
&aring;
&Aring;&ccedil;&euml;&Uuml; &ntilde;
&aring;−O
&Ccedil;&ntilde;
−
I &aring; ≠NK
&aring; −N
∫
(&aring; − N) &euml;&aacute;&aring;&Uuml; &ntilde; &aring; − N &euml;&aacute;&aring;&Uuml; &aring;−O &ntilde;
N
&aring; −N
&euml;&aacute;&aring;&Uuml; &ntilde; &Aring;&ccedil;&euml;&Uuml; &aring; −N &ntilde; &Aring;&ccedil;&euml;&Uuml; &ntilde; +
&Aring;&ccedil;&euml;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde;
∫
&aring;
&aring;
&euml;&aacute;&aring;&Uuml; &ntilde;
&aring;−O
&Ccedil;&ntilde;
+
I &aring; ≠NK
&aring; −N
∫
(&aring; − N) &Aring;&ccedil;&euml;&Uuml; &ntilde; &aring; − N &Aring;&ccedil;&euml;&Uuml; &aring;−O &ntilde;
&euml;&aacute;&aring;&Uuml; &aring; +N &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; −N &ntilde;
1024. ∫ &euml;&aacute;&aring;&Uuml; &ntilde; &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde; =
&aring;+&atilde;
&atilde; −N
+
&euml;&aacute;&aring;&Uuml; &aring; &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; −O &ntilde;&Ccedil;&ntilde;
∫
&aring;+&atilde;
&aring;
&atilde;
1025. ∫ &euml;&aacute;&aring;&Uuml; &aring; &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; &ntilde;&Ccedil;&ntilde; =
&euml;&aacute;&aring;&Uuml; &aring; −N &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; +N &ntilde;
&aring;+&atilde;
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CHAPTER 9. INTEGRAL CALCULUS
&aring; −N
&euml;&aacute;&aring;&Uuml; &aring; −O &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; &ntilde;&Ccedil;&ntilde;
∫
&aring;+&atilde;
−
1026. ∫ &iacute;~&aring;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; = −
N
&iacute;~&aring;&Uuml; &aring; −N &ntilde; + ∫ &iacute;~&aring;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde; I &aring; ≠ N K
&aring; −N
1027. ∫ &Aring;&ccedil;&iacute;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; = −
N
&Aring;&ccedil;&iacute;&Uuml; &aring; −N &ntilde; + ∫ &Aring;&ccedil;&iacute;&Uuml; &aring; − O &ntilde;&Ccedil;&ntilde; I &aring; ≠ N K
&aring; −N
&euml;&Eacute;&Aring;&Uuml; &aring; −O &ntilde; &iacute;~&aring;&Uuml; &ntilde; &aring; − O
1028. ∫ &euml;&Eacute;&Aring;&Uuml; &ntilde;&Ccedil;&ntilde; =
+
&euml;&Eacute;&Aring;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde; I &aring; ≠ N K
∫
&aring; −N
&aring; −N
&aring;
N
&aring; −N
1029. ∫ &euml;&aacute;&aring; &aring; &ntilde;&Ccedil;&ntilde; = − &euml;&aacute;&aring; &aring; −N &ntilde; &Aring;&ccedil;&euml; &ntilde; +
&euml;&aacute;&aring; &aring; −O &ntilde;&Ccedil;&ntilde;
∫
&aring;
&aring;
1030.
&Ccedil;&ntilde;
∫ &euml;&aacute;&aring;
&aring;
&ntilde;
=−
1031. ∫ &Aring;&ccedil;&euml; &aring; &ntilde;&Ccedil;&ntilde; =
1032.
&Ccedil;&ntilde;
∫ &Aring;&ccedil;&euml;
&aring;
&ntilde;
=
&Aring;&ccedil;&euml; &ntilde;
&aring;−O
&Ccedil;&ntilde;
+
I &aring; ≠NK
&aring; −N
∫
(&aring; − N) &euml;&aacute;&aring; &ntilde; &aring; − N &euml;&aacute;&aring; &aring;−O &ntilde;
&aring; −N
N
&euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &aring; −N &ntilde; +
&Aring;&ccedil;&euml; &aring; −O &ntilde;&Ccedil;&ntilde;
∫
&aring;
&aring;
&euml;&aacute;&aring; &ntilde;
&aring;−O
&Ccedil;&ntilde;
+
I &aring; ≠NK
&aring; −N
∫
(&aring; − N) &Aring;&ccedil;&euml; &ntilde; &aring; − N &Aring;&ccedil;&euml; &aring;−O &ntilde;
1033. ∫ &euml;&aacute;&aring; &aring; &ntilde; &Aring;&ccedil;&euml; &atilde; &ntilde;&Ccedil;&ntilde; =
+
&euml;&aacute;&aring; &aring;+N &ntilde; &Aring;&ccedil;&euml; &atilde; −N &ntilde;
&aring;+&atilde;
&atilde; −N
&euml;&aacute;&aring; &aring; &ntilde; &Aring;&ccedil;&euml; &atilde; −O &ntilde;&Ccedil;&ntilde;
&aring;+&atilde;∫
1034. ∫ &euml;&aacute;&aring; &aring; &ntilde; &Aring;&ccedil;&euml; &atilde; &ntilde;&Ccedil;&ntilde; = −
&euml;&aacute;&aring; &aring; −N &ntilde; &Aring;&ccedil;&euml; &atilde; +N &ntilde;
&aring;+&atilde;
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+
&aring; −N
&euml;&aacute;&aring; &aring; −O &ntilde; &Aring;&ccedil;&euml; &atilde; &ntilde;&Ccedil;&ntilde;
∫
&aring;+&atilde;
1035. ∫ &iacute;~&aring; &aring; &ntilde;&Ccedil;&ntilde; =
N
&iacute;~&aring; &aring; −N &ntilde; − ∫ &iacute;~&aring; &aring; −O &ntilde;&Ccedil;&ntilde; I &aring; ≠ N K
&aring; −N
1036. ∫ &Aring;&ccedil;&iacute; &aring; &ntilde;&Ccedil;&ntilde; = −
N
&Aring;&ccedil;&iacute; &aring; −N &ntilde; − ∫ &Aring;&ccedil;&iacute; &aring;−O &ntilde;&Ccedil;&ntilde; I &aring; ≠ N K
&aring; −N
&euml;&Eacute;&Aring; &aring; −O &ntilde; &iacute;~&aring; &ntilde; &aring; − O
1037. ∫ &euml;&Eacute;&Aring; &ntilde;&Ccedil;&ntilde; =
+
&euml;&Eacute;&Aring; &aring; −O &ntilde;&Ccedil;&ntilde; I &aring; ≠ N K
∫
&aring; −N
&aring; −N
&aring;
1038. ∫ &Aring;&euml;&Aring; &aring; &ntilde;&Ccedil;&ntilde; = −
&Aring;&euml;&Aring; &aring; −O &ntilde; &Aring;&ccedil;&iacute; &ntilde; &aring; − O
+
&Aring;&euml;&Aring; &aring; −O &ntilde;&Ccedil;&ntilde; I &aring; ≠ N K
∫
&aring; −N
&aring; −N
1039. ∫ &ntilde; &aring; &auml;&aring; &atilde; &ntilde;&Ccedil;&ntilde; =
1040.
&ntilde; &aring; +N &auml;&aring; &atilde; &ntilde; &atilde;
−
&ntilde; &aring; &auml;&aring; &atilde; −N &ntilde;&Ccedil;&ntilde;
&aring; +N
&aring; +N ∫
&auml;&aring; &atilde; &ntilde;
&auml;&aring; &atilde; &ntilde;
&atilde; &auml;&aring; &atilde; −N &ntilde;
&Ccedil;&ntilde;
=
−
+
&Ccedil;&ntilde; I &aring; ≠ N K
∫ &ntilde;&aring;
(&aring; − N)&ntilde; &aring;−N &aring; − N ∫ &ntilde; &aring;
1041. ∫ &auml;&aring; &aring; &ntilde;&Ccedil;&ntilde; = &ntilde; &auml;&aring; &aring; &ntilde; − &aring; ∫ &auml;&aring; &aring; −N &ntilde;&Ccedil;&ntilde;
1042. ∫ &ntilde; &aring; &euml;&aacute;&aring;&Uuml; &ntilde;&Ccedil;&ntilde; = &ntilde; &aring; &Aring;&ccedil;&euml;&Uuml; &ntilde; − &aring; ∫ &ntilde; &aring; −N &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde;
1043. ∫ &ntilde; &aring; &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde; = &ntilde; &aring; &euml;&aacute;&aring;&Uuml; &ntilde; − &aring; ∫ &ntilde; &aring; −N &euml;&aacute;&aring;&Uuml; &ntilde;&Ccedil;&ntilde;
1044. ∫ &ntilde; &aring; &euml;&aacute;&aring; &ntilde;&Ccedil;&ntilde; = − &ntilde; &aring; &Aring;&ccedil;&euml; &ntilde; + &aring; ∫ &ntilde; &aring; −N &Aring;&ccedil;&euml; &ntilde;&Ccedil;&ntilde;
1045. ∫ &ntilde; &aring; &Aring;&ccedil;&euml; &ntilde;&Ccedil;&ntilde; = &ntilde; &aring; &euml;&aacute;&aring; &ntilde; − &aring; ∫ &ntilde; &aring; −N &euml;&aacute;&aring; &ntilde;&Ccedil;&ntilde;
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1046. ∫ &ntilde; &aring; &euml;&aacute;&aring; −N &ntilde;&Ccedil;&ntilde; =
&ntilde; &aring; +N
N
&ntilde; &aring; +N
&euml;&aacute;&aring; −N &ntilde; −
&Ccedil;&ntilde;
&aring; +N
&aring; + N ∫ N− &ntilde;O
1047. ∫ &ntilde; &aring; &Aring;&ccedil;&euml; −N &ntilde;&Ccedil;&ntilde; =
&ntilde; &aring; +N
N
&ntilde; &aring; +N
&Aring;&ccedil;&euml; −N &ntilde; +
&Ccedil;&ntilde;
&aring; +N
&aring; + N ∫ N− &ntilde;O
1048. ∫ &ntilde; &aring; &iacute;~&aring; −N &ntilde;&Ccedil;&ntilde; =
N
&ntilde; &aring; +N
&ntilde; &aring; +N
&Ccedil;&ntilde;
&iacute;~&aring; −N &ntilde; −
&aring; +N
&aring; + N ∫ N+ &ntilde;O
&ntilde; &aring; &Ccedil;&ntilde;
&ntilde; &Auml;
&Ccedil;&ntilde;
1049. ∫ &aring;
= − ∫ &aring;
~&ntilde; + &Auml; ~ ~ ~&ntilde; + &Auml;
1050.
∫ (~&ntilde;
1051.
∫ (&ntilde;
1052.
&Ccedil;&ntilde;
&Ccedil;&ntilde;
+~
&aring; ≠NK
∫ (&ntilde;
O
&Ccedil;&ntilde;
O
− O~&ntilde; − &Auml;
=
(&aring; − N)(&Auml; − Q~&Aring; )(~&ntilde; O + &Auml;&ntilde; + &Aring; )
+ &Auml;&ntilde; + &Aring; )
O(O&aring; − P)~
&Ccedil;&ntilde;
−
I &aring; ≠NK
O
∫
O
(&aring; − N)(&Auml; − Q~&Aring; ) (~&ntilde; + &Auml;&ntilde; + &Aring; )&aring;−N
&aring;
O
)
=
)
=−
O &aring;
&aring; −N
O
&ntilde;
O(&aring; − N)~ O (&ntilde; O + ~
)
O &aring; −N
+
O&aring; − P
O(&aring; − N)~ O
&ntilde;
O(&aring; − N)~ O (&ntilde; O − ~ O )
O&aring; − P
&Ccedil;&ntilde;
−
I &aring; ≠NK
O ∫
O(&aring; − N)~ (&ntilde; O − ~ O )&aring;−N
−~
O &aring;
&aring; −N
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∫ (&ntilde;
&Ccedil;&ntilde;
O
+ ~O )
&aring; −N
I
CHAPTER 9. INTEGRAL CALCULUS
9.8 Definite Integral
&Auml;
&Auml;
~
~
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; I ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde; I &pound;
&aring;
o&aacute;&Eacute;&atilde;~&aring;&aring; &euml;&igrave;&atilde;W
∑ &Ntilde; (ξ )∆&ntilde;
&aacute;
&aacute; =N
&aacute;
p&atilde;~&auml;&auml; &Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;W ∆&ntilde; &aacute;
^&aring;&iacute;&aacute;&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;W c(&ntilde; ) I d(&ntilde; )
i&aacute;&atilde;&aacute;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;W ~I &Auml;I &Aring;I &Ccedil;
&Auml;
1053. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =
~
&aring;
&auml;&aacute;&atilde;
∑ &Ntilde; (ξ )∆&ntilde;
&aring;→∞
&atilde;~&ntilde; ∆&ntilde; &aacute; →M &aacute; =N
&aacute;
&aacute;
I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ∆&ntilde; &aacute; = &ntilde; &aacute; − &ntilde; &aacute; −N I &ntilde; &aacute; −N ≤ ξ &aacute; ≤ &ntilde; &aacute; K
Figure 179.
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&Auml;
1054. ∫ N&Ccedil;&ntilde; = &Auml; − ~
~
&Auml;
&Auml;
~
~
1055. ∫ &acirc;&Ntilde; (&ntilde; )&Ccedil;&ntilde; = &acirc; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde;
1056.
1057.
&Auml;
&Auml;
&Auml;
~
~
~
&Auml;
&Auml;
&Auml;
~
~
~
∫ [&Ntilde; (&ntilde; ) + &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde;
∫ [&Ntilde; (&ntilde; ) − &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; − ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde;
~
1058. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = M
~
&Auml;
~
~
&Auml;
1059. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = − ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde;
&Auml;
&Aring;
&Auml;
~
~
&Aring;
1060. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &Ntilde;&ccedil;&ecirc; ~ &lt; &Aring; &lt; &Auml; K
&Auml;
1061. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; ≥ M &aacute;&Ntilde; &Ntilde; (&ntilde; ) ≥ M &ccedil;&aring; [~ I &Auml;] K
~
&Auml;
1062. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; ≤ M &aacute;&Ntilde; &Ntilde; (&ntilde; ) ≤ M &ccedil;&aring; [~ I &Auml;] K
~
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CHAPTER 9. INTEGRAL CALCULUS
1063. c&igrave;&aring;&Ccedil;~&atilde;&Eacute;&aring;&iacute;~&auml; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde; &ccedil;&Ntilde; `~&auml;&Aring;&igrave;&auml;&igrave;&euml;
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = c(&ntilde; )
&Auml;
~
= c(&Auml;) − c(~ ) &aacute;&Ntilde; c′(&ntilde; ) = &Ntilde; (&ntilde; ) K
~
1064. j&Eacute;&iacute;&Uuml;&ccedil;&Ccedil; &ccedil;&Ntilde; p&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;
f&Ntilde; &ntilde; = &Ouml; (&iacute; ) I &iacute;&Uuml;&Eacute;&aring;
&Auml;
&Ccedil;
~
&Aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&Ouml;(&iacute; ))&Ouml;′(&iacute; )&Ccedil;&iacute; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&Aring; = &Ouml; −N (~ ) I &Ccedil; = &Ouml; −N (&Auml;) K
1065. f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring; &Auml;&oacute; m~&ecirc;&iacute;&euml;
&Auml;
&Auml;
∫ &igrave;&Ccedil;&icirc; = (&igrave;&icirc; ) ~ − ∫ &icirc;&Ccedil;&igrave;
~
&Auml;
~
1066. q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;~&auml; o&igrave;&auml;&Eacute;
&Auml;
&aring; −N
&Auml;−~ 

(
)
(
)
&Ntilde;
&ntilde;
&Ccedil;&ntilde;
&Ntilde;
(
&ntilde;
)
&Ntilde;
&ntilde;
O
&Ntilde; (&ntilde; &aacute; )
=
+
+
∑
M
&aring;

∫~
O&aring; 
&aacute; =N

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CHAPTER 9. INTEGRAL CALCULUS
Figure 180.
1067. p&aacute;&atilde;&eacute;&euml;&ccedil;&aring;∞&euml; o&igrave;&auml;&Eacute;
&Auml;
&Auml;−~
∫~ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = P&aring; [&Ntilde; (&ntilde; M ) + Q&Ntilde; (&ntilde; N ) + O&Ntilde; (&ntilde; O ) + Q&Ntilde; (&ntilde; P ) +
+ O&Ntilde; (&ntilde; Q ) + K + Q&Ntilde; (&ntilde; &aring; −N ) + &Ntilde; (&ntilde; &aring; )] I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&Auml;−~
&ntilde;&aacute; = ~ +
&aacute; I &aacute; = MI NI OI KI &aring; K
&aring;
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CHAPTER 9. INTEGRAL CALCULUS
Figure 181.
1068. ^&ecirc;&Eacute;~ r&aring;&Ccedil;&Eacute;&ecirc; ~ `&igrave;&ecirc;&icirc;&Eacute;
&Auml;
p = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = c(&Auml;) − c(~ ) I
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; c′(&ntilde; ) = &Ntilde; (&ntilde; ) K
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CHAPTER 9. INTEGRAL CALCULUS
Figure 182.
1069. ^&ecirc;&Eacute;~ _&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; q&iuml;&ccedil; `&igrave;&ecirc;&icirc;&Eacute;&euml;
&Auml;
p = ∫ [&Ntilde; (&ntilde; ) − &Ouml; (&ntilde; )]&Ccedil;&ntilde; = c(&Auml;) − d(&Auml;) − c(~ ) + d(~ ) I
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; c′(&ntilde; ) = &Ntilde; (&ntilde; ) I d′(&ntilde; ) = &Ouml; (&ntilde; ) K
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CHAPTER 9. INTEGRAL CALCULUS
Figure 183.
9.9 Improper Integral
&Auml;
1070. q&Uuml;&Eacute; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &aacute;&euml; &Aring;~&auml;&auml;&Eacute;&Ccedil; ~&aring; &aacute;&atilde;&eacute;&ecirc;&ccedil;&eacute;&Eacute;&ecirc; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;
~
&aacute;&Ntilde;
•
•
~ &ccedil;&ecirc; &Auml; &aacute;&euml; &aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;I
&Ntilde; (&ntilde; ) &Uuml;~&euml; &ccedil;&aring;&Eacute; &ccedil;&ecirc; &atilde;&ccedil;&ecirc;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &Ccedil;&aacute;&euml;&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&aacute;&iacute;&oacute;
&aacute;&aring; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; [~ I &Auml;] K
1071. f&Ntilde; &Ntilde; (&ntilde; ) &aacute;&euml; ~ &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&aring; [~ I ∞ ) I &iacute;&Uuml;&Eacute;&aring;
∞
&aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; K
~
&aring; →∞
~
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CHAPTER 9. INTEGRAL CALCULUS
Figure 184.
1072. f&Ntilde; &Ntilde; (&ntilde; ) &aacute;&euml; ~ &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&aring; (− ∞I &Auml;] I &iacute;&Uuml;&Eacute;&aring;
&Auml;
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; K
−∞
&aring;→ −∞
&aring;
Figure 185.
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k&ccedil;&iacute;&Eacute; W q&Uuml;&Eacute; &aacute;&atilde;&eacute;&ecirc;&ccedil;&eacute;&Eacute;&ecirc; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &aacute;&aring; NMTNI NMTO ~&ecirc;&Eacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;
&aacute;&Ntilde; &iacute;&Uuml;&Eacute; &auml;&aacute;&atilde;&aacute;&iacute;&euml; &Eacute;&ntilde;&aacute;&euml;&iacute; ~&aring;&Ccedil; ~&ecirc;&Eacute; &Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;X &ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; ~&ecirc;&Eacute;
&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
1073.
∞
&Aring;
∞
−∞
−∞
&Aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde;
Figure 186.
f&Ntilde; &Ntilde;&ccedil;&ecirc; &euml;&ccedil;&atilde;&Eacute; &ecirc;&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &Aring;I &Auml;&ccedil;&iacute;&Uuml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &ecirc;&aacute;&Ouml;&Uuml;&iacute;
∞
&euml;&aacute;&Ccedil;&Eacute; ~&ecirc;&Eacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde;
&aacute;&euml; ~&auml;&euml;&ccedil;
−∞
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;X &ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute; &aacute;&iacute; &aacute;&euml; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
1074. `&ccedil;&atilde;&eacute;~&ecirc;&aacute;&euml;&ccedil;&aring; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;&euml;
i&Eacute;&iacute; &Ntilde; (&ntilde; ) ~&aring;&Ccedil; &Ouml;(&ntilde; ) &Auml;&Eacute; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;
&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; [~ I ∞ ) K p&igrave;&eacute;&eacute;&ccedil;&euml;&Eacute; &iacute;&Uuml;~&iacute; M ≤ &Ouml; (&ntilde; ) ≤ &Ntilde; (&ntilde; ) &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; &aacute;&aring;
[~I ∞ ) K
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•
∞
∞
~
~
f&Ntilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I &iacute;&Uuml;&Eacute;&aring; ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde; &aacute;&euml; ~&auml;&euml;&ccedil;
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I
∞
•
f&Ntilde; ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde;
~
∞
&aacute;&euml; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I &iacute;&Uuml;&Eacute;&aring; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &aacute;&euml; ~&auml;&euml;&ccedil; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
~
1075. ^&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;
f&Ntilde;
∞
∞
~
~
∫ &Ntilde; (&ntilde; ) &Ccedil;&ntilde; &aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &aacute;&euml; ~&Auml;&euml;&ccedil;-
&auml;&igrave;&iacute;&Eacute;&auml;&oacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
1076. a&aacute;&euml;&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&aring;&Ccedil;
i&Eacute;&iacute; &Ntilde; (&ntilde; ) &Auml;&Eacute; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &iuml;&Uuml;&aacute;&Aring;&Uuml; &aacute;&euml; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;
[~I &Auml;) &Auml;&igrave;&iacute; &aacute;&euml; &Ccedil;&aacute;&euml;&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; ~&iacute; &ntilde; = &Auml; K q&Uuml;&Eacute;&aring;
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde;
~
ε →M +
&Auml;−ε
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde;
~
Figure 187.
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1077. i&Eacute;&iacute; &Ntilde; (&ntilde; ) &Auml;&Eacute; ~ &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ecirc;&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; &ntilde; &aacute;&aring;
&iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; [~ I &Auml;] &Eacute;&ntilde;&Aring;&Eacute;&eacute;&iacute; &Ntilde;&ccedil;&ecirc; &euml;&ccedil;&atilde;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; &Aring; &aacute;&aring; (~ I &Auml;) K q&Uuml;&Eacute;&aring;
&Aring; −ε
&Auml;
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; K
~
ε →M +
δ →M +
~
&Aring; +δ
Figure 188.
9.10 Double Integral
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&iuml;&ccedil; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &Ntilde; (&ntilde; I &oacute; ) I &Ntilde; (&igrave;I &icirc; ) I &pound;
a&ccedil;&igrave;&Auml;&auml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;W
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; I ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; I &pound;
o
o&aacute;&Eacute;&atilde;~&aring;&aring; &euml;&igrave;&atilde;W
o
∑∑ &Ntilde; (&igrave; I &icirc; )∆&ntilde; ∆&oacute;
&atilde;
&aring;
&aacute; =N &agrave;=N
&aacute;
&agrave;
&aacute;
&agrave;
p&atilde;~&auml;&auml; &Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;W ∆&ntilde; &aacute; I ∆&oacute; &agrave;
o&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;W oI p
m&ccedil;&auml;~&ecirc; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ecirc; I θ
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^&ecirc;&Eacute;~W ^
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W p
s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ &euml;&ccedil;&auml;&aacute;&Ccedil;W s
j~&euml;&euml; &ccedil;&Ntilde; ~ &auml;~&atilde;&aacute;&aring;~W &atilde;
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W ρ(&ntilde; I &oacute; )
c&aacute;&ecirc;&euml;&iacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W j &ntilde; I j &oacute;
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W f &ntilde; I f &oacute; I fM
`&Uuml;~&ecirc;&Ouml;&Eacute; &ccedil;&Ntilde; ~ &eacute;&auml;~&iacute;&Eacute;W n
`&Uuml;~&ecirc;&Ouml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W σ(&ntilde; I &oacute; )
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &atilde;~&euml;&euml;W &ntilde; I &oacute;
^&icirc;&Eacute;&ecirc;~&Ouml;&Eacute; &ccedil;&Ntilde; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W &micro;
1078. a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; a&ccedil;&igrave;&Auml;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;
q&Uuml;&Eacute; &Ccedil;&ccedil;&igrave;&Auml;&auml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&icirc;&Eacute;&ecirc; ~ &ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute; [~I &Auml;]&times; [&Aring;I &Ccedil;] &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;
&iacute;&ccedil; &Auml;&Eacute;
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = &auml;&aacute;&atilde;
[~ I &Auml; ]&times;[&Aring; I &Ccedil; ]
(
)
∑∑ &Ntilde; (&igrave; I &icirc; )∆&ntilde; ∆&oacute;
&atilde;
&aring;
&atilde;~&ntilde; ∆&ntilde; &aacute; →M
&atilde;~&ntilde; ∆&oacute; &agrave; → M &aacute; =N &agrave;=N
&aacute;
&agrave;
&aacute;
&agrave;
I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &igrave; &aacute; I &icirc; &agrave; &aacute;&euml; &euml;&ccedil;&atilde;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; &aacute;&aring; &iacute;&Uuml;&Eacute; &ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;
(&ntilde; &aacute; −N I &ntilde; &aacute; )&times; &oacute; &agrave;−N I &oacute; &agrave; I ~&aring;&Ccedil; ∆&ntilde; &aacute; = &ntilde; &aacute; − &ntilde; &aacute; −N I ∆&oacute; &agrave; = &oacute; &agrave; − &oacute; &agrave;−N K
(
)
Figure 189.
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CHAPTER 9. INTEGRAL CALCULUS
q&Uuml;&Eacute; &Ccedil;&ccedil;&igrave;&Auml;&auml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&icirc;&Eacute;&ecirc; ~ &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; o &aacute;&euml;
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;^ I
o
[~ I &Auml; ]&times;[&Aring; I &Ccedil; ]
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute; [~I &Auml;]&times; [&Aring;I &Ccedil;] &Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&euml; oI
&Ouml;(&ntilde; I &oacute; ) = &Ntilde; (&ntilde; I &oacute; ) &aacute;&Ntilde; &Ntilde; (&ntilde; I &oacute; ) &aacute;&euml; &aacute;&aring; o ~&aring;&Ccedil; &Ouml;(&ntilde; I &oacute; ) = M &ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute;K
Figure 190.
1079.
∫∫ [&Ntilde; (&ntilde; I &oacute; ) + &Ouml;(&ntilde; I &oacute; )]&Ccedil;^ = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ + ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;^
o
1080.
o
∫∫ [&Ntilde; (&ntilde; I &oacute; ) − &Ouml;(&ntilde; I &oacute; )]&Ccedil;^ = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ − ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;^
o
1081.
o
o
o
∫∫ &acirc;&Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = &acirc; ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ I
o
o
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &acirc; &aacute;&euml; ~ &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K
1082. f&Ntilde; &Ntilde; (&ntilde; I &oacute; ) ≤ &Ouml;(&ntilde; I &oacute; ) &ccedil;&aring; oI &iacute;&Uuml;&Eacute;&aring;
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ ≤ ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;^ K
o
1083. f&Ntilde; &Ntilde; (&ntilde; I &oacute; ) ≥ M &ccedil;&aring; o ~&aring;&Ccedil; p ⊂ o I &iacute;&Uuml;&Eacute;&aring;
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o
CHAPTER 9. INTEGRAL CALCULUS
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ ≤ ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ K
p
o
Figure 191.
1084. f&Ntilde; &Ntilde; (&ntilde; I &oacute; ) ≥ M &ccedil;&aring; o ~&aring;&Ccedil; o ~&aring;&Ccedil; p ~&ecirc;&Eacute; &aring;&ccedil;&aring;-&ccedil;&icirc;&Eacute;&ecirc;&auml;~&eacute;&eacute;&aacute;&aring;&Ouml;
&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;I &iacute;&Uuml;&Eacute;&aring;
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ + ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ K
o ∪p
o
p
e&Eacute;&ecirc;&Eacute; o ∪ p &aacute;&euml; &iacute;&Uuml;&Eacute; &igrave;&aring;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml; o ~&aring;&Ccedil; pK
Figure 192.
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CHAPTER 9. INTEGRAL CALCULUS
1085. f&iacute;&Eacute;&ecirc;~&iacute;&Eacute;&Ccedil; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; ~&aring;&Ccedil; c&igrave;&Auml;&aacute;&aring;&aacute;∞&euml; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;
&Auml; &egrave;(&ntilde; )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫( &Ntilde;) (&ntilde; I &oacute; )&Ccedil;&oacute;&Ccedil;&ntilde;
o
~&eacute; &ntilde;
&Ntilde;&ccedil;&ecirc; ~ &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&oacute;&eacute;&Eacute; fI
o = {(&ntilde; I &oacute; ) &ouml; ~ ≤ &ntilde; ≤ &Auml;I &eacute;(&ntilde; ) ≤ &oacute; ≤ &egrave;(&ntilde; )} K
Figure 193.
&Ccedil; &icirc; (&oacute; )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute;
o
&Aring; &igrave;(&oacute; )
&Ntilde;&ccedil;&ecirc; ~ &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&oacute;&eacute;&Eacute; ffI
o = {(&ntilde; I &oacute; ) &ouml; &igrave;(&oacute; ) ≤ &ntilde; ≤ &icirc; (&oacute; )I &Aring; ≤ &oacute; ≤ &Ccedil;} K
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CHAPTER 9. INTEGRAL CALCULUS
Figure 194.
1086. a&ccedil;&igrave;&Auml;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&icirc;&Eacute;&ecirc; o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc; o&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;
f&Ntilde; o &aacute;&euml; &iacute;&Uuml;&Eacute; &ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc; &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; [~I &Auml;]&times; [&Aring;I &Ccedil;] I &iacute;&Uuml;&Eacute;&aring;
&Auml; &Ccedil;
&Ccedil; &Auml;





 ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde; &Ccedil;&oacute; K

(
)
(
)
&Ntilde;
&ntilde;
I
&oacute;
&Ccedil;&ntilde;&Ccedil;&oacute;
&Ntilde;
&ntilde;
I
&oacute;
&Ccedil;&oacute;
&Ccedil;&ntilde;
=
=
∫∫o
∫~  ∫&Aring;
∫



&Aring;~


f&aring; &iacute;&Uuml;&Eacute; &euml;&eacute;&Eacute;&Aring;&aacute;~&auml; &Aring;~&euml;&Eacute; &iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&aring;&Ccedil; &Ntilde; (&ntilde; I &oacute; ) &Aring;~&aring; &Auml;&Eacute; &iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring; ~&euml; &Ouml; (&ntilde; )&Uuml;(&oacute; ) &iuml;&Eacute; &Uuml;~&icirc;&Eacute;

&Ccedil;
&Auml;
K



(
)
(
)
(
)
(
)
(
)
&Ntilde;
&ntilde;
I
&oacute;
&Ccedil;&ntilde;&Ccedil;&oacute;
&Ouml;
&ntilde;
&Uuml;
&oacute;
&Ccedil;&ntilde;&Ccedil;&oacute;
&Ouml;
&ntilde;
&Ccedil;&ntilde;
&Uuml;
&oacute;
&Ccedil;&oacute;
=
=
∫∫o
∫∫o

∫
∫

&Aring;
~
1087. `&Uuml;~&aring;&Ouml;&Eacute; &ccedil;&Ntilde; s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;
∂ (&ntilde; I &oacute; )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; = ∫∫ &Ntilde; [&ntilde; (&igrave;I &icirc; )I &oacute;(&igrave;I &icirc; )] ∂(&igrave;I &icirc; ) &Ccedil;&igrave;&Ccedil;&icirc; I
o
p
∂&ntilde; ∂&ntilde;
∂ (&ntilde; I &oacute; ) ∂&igrave; ∂&icirc;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
=
≠ M &aacute;&euml; &iacute;&Uuml;&Eacute; &agrave;~&Aring;&ccedil;&Auml;&aacute;~&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &iacute;&ecirc;~&aring;&euml;∂ (&igrave;I &icirc; ) ∂&oacute; ∂&oacute;
∂&igrave; ∂&icirc;
&Ntilde;&ccedil;&ecirc;&atilde;~&iacute;&aacute;&ccedil;&aring;&euml; (&ntilde; I &oacute; ) → (&igrave;I &icirc; ) I ~&aring;&Ccedil; p &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc; &ccedil;&Ntilde; o &iuml;&Uuml;&aacute;&Aring;&Uuml;
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CHAPTER 9. INTEGRAL CALCULUS
&Aring;~&aring; &Auml;&Eacute; &Aring;&ccedil;&atilde;&eacute;&igrave;&iacute;&Eacute;&Ccedil; &Auml;&oacute; &ntilde; = &ntilde; (&igrave;I &icirc; ) I &oacute; = &oacute; (&igrave;I &icirc; ) &aacute;&aring;&iacute;&ccedil; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; oK
1088. m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
&ntilde; = &ecirc; &Aring;&ccedil;&euml; θ I &oacute; = &ecirc; &euml;&aacute;&aring; θ K
Figure 195.
1089. a&ccedil;&igrave;&Auml;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &aacute;&aring; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
q&Uuml;&Eacute; a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Ccedil;&ntilde;&Ccedil;&oacute; &Ntilde;&ccedil;&ecirc; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &aacute;&euml;
∂ (&ntilde; I &oacute; )
&Ccedil;&ntilde;&Ccedil;&oacute; =
&Ccedil;&ecirc;&Ccedil;θ = &ecirc;&Ccedil;&ecirc;&Ccedil;θ K
∂ (&ecirc;I θ)
i&Eacute;&iacute; &iacute;&Uuml;&Eacute; &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; o &aacute;&euml; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&Eacute;&Ccedil; ~&euml; &Ntilde;&ccedil;&auml;&auml;&ccedil;&iuml;&euml;W
M ≤ &Ouml; (θ) ≤ &ecirc; ≤ &Uuml;(θ) I α ≤ θ ≤ β I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; β − α ≤ Oπ K
q&Uuml;&Eacute;&aring;
β &Uuml; (θ )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; = ∫ ∫ &Ntilde; (&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θ)&ecirc;&Ccedil;&ecirc;&Ccedil;θ K
o
α &Ouml; (θ )
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CHAPTER 9. INTEGRAL CALCULUS
Figure 196.
f&Ntilde; &iacute;&Uuml;&Eacute; &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; o &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&auml;~&ecirc; &ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
M ≤ ~ ≤ &ecirc; ≤ &Auml; I α ≤ θ ≤ β I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; β − α ≤ Oπ I
&iacute;&Uuml;&Eacute;&aring;
β &Auml;
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; = ∫ ∫ &Ntilde; (&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θ)&ecirc;&Ccedil;&ecirc;&Ccedil;θ K
o
α ~
Figure 197.
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CHAPTER 9. INTEGRAL CALCULUS
1090. ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ o&Eacute;&Ouml;&aacute;&ccedil;&aring;
&Auml; &Ntilde; (&ntilde; )
^ = ∫ ∫ &Ccedil;&oacute;&Ccedil;&ntilde; E&Ntilde;&ccedil;&ecirc; ~ &iacute;&oacute;&eacute;&Eacute; f &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;FK
~ &Ouml; (&ntilde; )
Figure 198.
&Ccedil; &egrave;(&oacute; )
^ = ∫ ∫ &Ccedil;&ntilde;&Ccedil;&oacute; E&Ntilde;&ccedil;&ecirc; ~ &iacute;&oacute;&eacute;&Eacute; ff &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;FK
&Aring; &eacute;( &oacute; )
Figure 199.
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1091. s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ p&ccedil;&auml;&aacute;&Ccedil;
s = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ K
o
Figure 200.
f&Ntilde; o &aacute;&euml; ~ &iacute;&oacute;&eacute;&Eacute; f &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; &Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; &ntilde; = ~ I &ntilde; = &Auml; I &oacute; = &Uuml;(&ntilde; ) I
&oacute; = &Ouml; (&ntilde; ) I &iacute;&Uuml;&Eacute;&aring;
&Auml; &Ouml;(&ntilde; )
s = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&oacute;&Ccedil;&ntilde; K
o
~ &Uuml;(&ntilde; )
f&Ntilde; o &aacute;&euml; ~ &iacute;&oacute;&eacute;&Eacute; ff &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; &Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; &oacute; = &Aring; I &oacute; = &Ccedil; I &ntilde; = &egrave;(&oacute; ) I
&ntilde; = &eacute;(&oacute; ) I &iacute;&Uuml;&Eacute;&aring;
&Ccedil; &egrave;(&oacute; )
s = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; K
o
&Aring; &eacute;( &oacute; )
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f&Ntilde; &Ntilde; (&ntilde; I &oacute; ) ≥ &Ouml; (&ntilde; I &oacute; ) &ccedil;&icirc;&Eacute;&ecirc; ~ &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; oI &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &icirc;&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde;
&iacute;&Uuml;&Eacute; &euml;&ccedil;&auml;&aacute;&Ccedil; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &ograve;N = &Ntilde; (&ntilde; I &oacute; ) ~&aring;&Ccedil; &ograve; O = &Ouml;(&ntilde; I &oacute; ) &ccedil;&icirc;&Eacute;&ecirc; o &aacute;&euml;
&Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
s = ∫∫ [&Ntilde; (&ntilde; I &oacute; ) − &Ouml; (&ntilde; I &oacute; )]&Ccedil;^ K
o
1092. ^&ecirc;&Eacute;~ ~&aring;&Ccedil; s&ccedil;&auml;&igrave;&atilde;&Eacute; &aacute;&aring; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
f&Ntilde; p &aacute;&euml; ~ &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; &aacute;&aring; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute; &Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; θ = α I θ = β I
&ecirc; = &Uuml;(θ) I &ecirc; = &Ouml; (θ) I
&iacute;&Uuml;&Eacute;&aring;
β &Ouml; (θ )
^ = ∫∫ &Ccedil;^ = ∫ ∫ &ecirc;&Ccedil;&ecirc;&Ccedil;θ I
p
α &Uuml;(θ )
s = ∫∫ &Ntilde; (&ecirc;I θ)&ecirc;&Ccedil;&ecirc;&Ccedil;θ K
p
Figure 201.
1093. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ^&ecirc;&Eacute;~
p = ∫∫
o
O
 ∂&ograve;   ∂&ograve; 
N +   +   &Ccedil;&ntilde;&Ccedil;&oacute;
 ∂&ntilde;   ∂&oacute; 
O
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1094. j~&euml;&euml; &ccedil;&Ntilde; ~ i~&atilde;&aacute;&aring;~
&atilde; = ∫∫ ρ(&ntilde; I &oacute; )&Ccedil;^ I
o
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &auml;~&atilde;&aacute;&aring;~ &ccedil;&Aring;&Aring;&igrave;&eacute;&aacute;&Eacute;&euml; ~ &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; o ~&aring;&Ccedil; &aacute;&iacute;&euml; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; ~&iacute;
~ &eacute;&ccedil;&aacute;&aring;&iacute; E&ntilde;I&oacute;F &aacute;&euml; ρ(&ntilde; I &oacute; ) K
1095. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;
q&Uuml;&Eacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;~&atilde;&aacute;&aring;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~
j &ntilde; = ∫∫ &oacute;ρ(&ntilde; I &oacute; )&Ccedil;^ K
o
q&Uuml;&Eacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &auml;~&atilde;&aacute;&aring;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &oacute;-~&ntilde;&aacute;&euml; &aacute;&euml;
j &oacute; = ∫∫ &ntilde;ρ(&ntilde; I &oacute; )&Ccedil;^ K
o
q&Uuml;&Eacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml; &aacute;&euml;
f &ntilde; = ∫∫ &oacute; Oρ(&ntilde; I &oacute; )&Ccedil;^ K
o
q&Uuml;&Eacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &oacute;-~&ntilde;&aacute;&euml; &aacute;&euml;
f &oacute; = ∫∫ &ntilde; Oρ(&ntilde; I &oacute; )&Ccedil;^ K
o
q&Uuml;&Eacute; &eacute;&ccedil;&auml;~&ecirc; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~ &aacute;&euml;
fM = ∫∫ (&ntilde; O + &oacute; O )ρ(&ntilde; I &oacute; )&Ccedil;^ K
o
1096. `&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; j~&euml;&euml;
j&oacute;
N
= ∫∫ &ntilde;ρ(&ntilde; I &oacute; )&Ccedil;^ =
&ntilde;=
&atilde; &atilde; o
∫∫ &ntilde;ρ(&ntilde; I &oacute; )&Ccedil;^
o
∫∫ ρ(&ntilde; I &oacute; )&Ccedil;^
o
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CHAPTER 9. INTEGRAL CALCULUS
∫∫ &oacute;ρ(&ntilde; I &oacute; )&Ccedil;^
j
N
&oacute; = &ntilde; = ∫∫ &oacute;ρ(&ntilde; I &oacute; )&Ccedil;^ =
&atilde; &atilde; o
o
∫∫ ρ(&ntilde; I &oacute; )&Ccedil;^
K
o
1097. `&Uuml;~&ecirc;&Ouml;&Eacute; &ccedil;&Ntilde; ~ m&auml;~&iacute;&Eacute;
n = ∫∫ σ(&ntilde; I &oacute; )&Ccedil;^ I
o
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;~&auml; &Aring;&Uuml;~&ecirc;&Ouml;&Eacute; &aacute;&euml; &Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&Eacute;&Ccedil; &ccedil;&icirc;&Eacute;&ecirc; ~ &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; o ~&aring;&Ccedil; &aacute;&iacute;&euml;
&Aring;&Uuml;~&ecirc;&Ouml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; ~&iacute; ~ &eacute;&ccedil;&aacute;&aring;&iacute; E&ntilde;I&oacute;F &aacute;&euml; σ(&ntilde; I &oacute; ) K
1098. ^&icirc;&Eacute;&ecirc;~&Ouml;&Eacute; &ccedil;&Ntilde; ~ c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
N
&micro; = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ I
po
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; p = ∫∫ &Ccedil;^ K
o
9.11 Triple Integral
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&ecirc;&Eacute;&Eacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &Ntilde; (&ntilde; I &oacute; I &ograve; ) I &Ouml; (&ntilde; I &oacute; I &ograve; ) I &pound;
q&ecirc;&aacute;&eacute;&auml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;W
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s I ∫∫∫ &Ouml;(&ntilde; I &oacute; I &ograve; )&Ccedil;s I &pound;
d
o&aacute;&Eacute;&atilde;~&aring;&aring; &euml;&igrave;&atilde;W
d
∑∑∑ &Ntilde; (&igrave; I &icirc; I &iuml; )∆&ntilde; ∆&oacute; ∆&ograve;
&atilde;
&aring;
&eacute;
&aacute; =N &agrave;=N &acirc; =N
&aacute;
&agrave;
&acirc;
&aacute;
p&atilde;~&auml;&auml; &Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;W ∆&ntilde; &aacute; I ∆&oacute; &agrave; I ∆&ograve; &acirc;
i&aacute;&atilde;&aacute;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;W ~I &Auml;I &Aring;I &Ccedil;I &ecirc;I &euml;
o&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;W dI qI p
`&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ecirc; I θ I &ograve;
p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ecirc; I θ I ϕ
s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ &euml;&ccedil;&auml;&aacute;&Ccedil;W s
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&acirc;
CHAPTER 9. INTEGRAL CALCULUS
j~&euml;&euml; &ccedil;&Ntilde; ~ &euml;&ccedil;&auml;&aacute;&Ccedil;W &atilde;
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W &micro;(&ntilde; I &oacute; I &ograve; )
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &atilde;~&euml;&euml;W &ntilde; I &oacute; I &ograve;
c&aacute;&ecirc;&euml;&iacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W j &ntilde;&oacute; I j &oacute;&ograve; I j &ntilde;&ograve;
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W f &ntilde;&oacute; I f &oacute;&ograve; I f &ntilde;&ograve; I f &ntilde; I f &oacute; I f&ograve; I fM
1099. a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; q&ecirc;&aacute;&eacute;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;
q&Uuml;&Eacute; &iacute;&ecirc;&aacute;&eacute;&auml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&icirc;&Eacute;&ecirc; ~ &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil; [~I &Auml;]&times; [&Aring;I &Ccedil;]&times; [&ecirc;I &euml;]
&aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; &iacute;&ccedil; &Auml;&Eacute;
∫∫∫
&Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;s =
[~ I &Auml; ]&times;[&Aring; I &Ccedil; ]&times;[&ecirc; I &euml; ]
(
)
)
∑∑∑ &Ntilde; (&igrave; I &icirc; I &iuml; )∆&ntilde; ∆&oacute; ∆&ograve;
&atilde;
&auml;&aacute;&atilde;
&eacute;
&aring;
&atilde;~&ntilde; ∆&ntilde; &aacute; →M
&atilde;~&ntilde; ∆&oacute; &agrave; →M &aacute; =N &agrave;=N &acirc; =N
&atilde;~&ntilde; ∆&ograve; &acirc; →M
&aacute;
&agrave;
&acirc;
&aacute;
&agrave;
&acirc;
I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &igrave; &aacute; I &icirc; &agrave; I &iuml; &acirc; &aacute;&euml; &euml;&ccedil;&atilde;&Eacute; &eacute;&ccedil;&aacute;&aring;&iacute; &aacute;&aring; &iacute;&Uuml;&Eacute; &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;
(&ntilde; &aacute; −N I &ntilde; &aacute; )&times; &oacute; &agrave;−N I &oacute; &agrave; &times; (&ograve; &acirc; −N I &ograve; &acirc; ) I ~&aring;&Ccedil; ∆&ntilde; &aacute; = &ntilde; &aacute; − &ntilde; &aacute; −N I
∆&oacute; &agrave; = &oacute; &agrave; − &oacute; &agrave;−N I ∆&ograve; &acirc; = &ograve; &acirc; − &ograve; &acirc; −N K
1100.
(
∫∫∫ [&Ntilde; (&ntilde; I &oacute;I &ograve; ) + &Ouml;(&ntilde; I &oacute;I &ograve; )]&Ccedil;s = ∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s + ∫∫∫ &Ouml;(&ntilde; I &oacute;I &ograve; )&Ccedil;s
d
1101.
d
∫∫∫ [&Ntilde; (&ntilde; I &oacute; I &ograve; ) − &Ouml;(&ntilde; I &oacute;I &ograve; )]&Ccedil;s = ∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s − ∫∫∫ &Ouml;(&ntilde; I &oacute;I &ograve; )&Ccedil;s
d
1102.
d
d
d
∫∫∫ &acirc;&Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s = &acirc; ∫∫∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;s I
d
d
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &acirc; &aacute;&euml; ~ &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K
1103. f&Ntilde; &Ntilde; (&ntilde; I &oacute; I &ograve; ) ≥ M ~&aring;&Ccedil; d ~&aring;&Ccedil; q ~&ecirc;&Eacute; &aring;&ccedil;&aring;&ccedil;&icirc;&Eacute;&ecirc;&auml;~&eacute;&eacute;&aacute;&aring;&Ouml; &Auml;~&euml;&aacute;&Aring;
&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;I &iacute;&Uuml;&Eacute;&aring;
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s = ∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s + ∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s K
d∪q
d
q
e&Eacute;&ecirc;&Eacute; d ∪ q &aacute;&euml; &iacute;&Uuml;&Eacute; &igrave;&aring;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml; d ~&aring;&Ccedil; qK
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1104. b&icirc;~&auml;&igrave;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; q&ecirc;&aacute;&eacute;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &Auml;&oacute; o&Eacute;&eacute;&Eacute;~&iacute;&Eacute;&Ccedil; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;
f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&ccedil;&auml;&aacute;&Ccedil; d &aacute;&euml; &iacute;&Uuml;&Eacute; &euml;&Eacute;&iacute; &ccedil;&Ntilde; &eacute;&ccedil;&aacute;&aring;&iacute;&euml; (&ntilde; I &oacute; I &ograve; ) &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute;
(&ntilde; I &oacute; )∈ oI χN (&ntilde; I &oacute; ) ≤ &ograve; ≤ χ O (&ntilde; I &oacute; ) I &iacute;&Uuml;&Eacute;&aring;

χ O ( &ntilde; I &oacute; )
(
)
(
)
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ograve;
=
 &Ccedil;&ntilde;&Ccedil;&oacute; I

∫∫∫
∫∫o  χ (∫&ntilde; I&oacute; )

d
N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; o &aacute;&euml; &eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; d &ccedil;&aring;&iacute;&ccedil; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K
f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&ccedil;&auml;&aacute;&Ccedil; d &aacute;&euml; &iacute;&Uuml;&Eacute; &euml;&Eacute;&iacute; &ccedil;&Ntilde; &eacute;&ccedil;&aacute;&aring;&iacute;&euml; (&ntilde; I &oacute; I &ograve; ) &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute;
~ ≤ &ntilde; ≤ &Auml;I ϕN (&ntilde; ) ≤ &oacute; ≤ ϕO (&ntilde; )I χN (&ntilde; I &oacute; ) ≤ &ograve; ≤ χ O (&ntilde; I &oacute; ) I &iacute;&Uuml;&Eacute;&aring;
∫∫∫
d
 ϕO ( &ntilde; )  χ O ( &ntilde; I &oacute; )
 
&Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; = ∫  ∫  ∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;&ograve; &Ccedil;&oacute; &Ccedil;&ntilde;
 

~ 
 
 ϕN ( &ntilde; )  χN ( &ntilde; I &oacute; )
&Auml;
1105. q&ecirc;&aacute;&eacute;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&icirc;&Eacute;&ecirc; m~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;
f&Ntilde; d &aacute;&euml; ~ &eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil; [~ I &Auml;]&times; [&Aring;I &Ccedil;]&times; [&ecirc;I &euml;] I &iacute;&Uuml;&Eacute;&aring;
&Auml; &Ccedil; &euml;
 
 

&Ccedil;&oacute; &Ccedil;&ntilde; K
(
)
(
)
=
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ograve;

∫∫∫
∫
∫
∫


d
~ &Aring;  &ecirc;
 
f&aring; &iacute;&Uuml;&Eacute; &euml;&eacute;&Eacute;&Aring;&aacute;~&auml; &Aring;~&euml;&Eacute; &iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&aring;&Ccedil; &Ntilde; (&ntilde; I &oacute; I &ograve; ) &Aring;~&aring; &Auml;&Eacute;
&iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring; ~&euml; &Ouml; (&ntilde; ) &Uuml;(&oacute; ) &acirc; (&ograve; ) &iuml;&Eacute; &Uuml;~&icirc;&Eacute;
&Auml;
 &Ccedil;
 &euml;




 ∫ &acirc; (&ograve; )&Ccedil;&ograve;  K
(
)
(
)
(
)
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;
&Ouml;
&ntilde;
&Ccedil;&ntilde;
&Uuml;
&oacute;
&Ccedil;&oacute;
=
∫∫∫

∫
 ∫

d

~
 &Aring;
 &ecirc;
1106. `&Uuml;~&aring;&Ouml;&Eacute; &ccedil;&Ntilde; s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; =
d
= ∫∫∫ &Ntilde; [&ntilde; (&igrave;I &icirc; I &iuml; )I &oacute; (&igrave;I &icirc; I &iuml; )I &ograve; (&igrave;I &icirc; I &iuml; )]
p
∂ (&ntilde; I &oacute; I &ograve; )
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;I
∂(&igrave;I &icirc; I &iuml; )
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∂&ntilde; ∂&ntilde; ∂&ntilde;
∂&igrave; ∂&icirc; ∂&iuml;
∂(&ntilde; I &oacute; I &ograve; ) ∂&oacute; ∂&oacute; ∂&oacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
=
≠ M &aacute;&euml; &iacute;&Uuml;&Eacute; &agrave;~&Aring;&ccedil;&Auml;&aacute;~&aring; &ccedil;&Ntilde;
∂(&igrave;I &icirc; I &iuml; ) ∂&igrave; ∂&icirc; ∂&iuml;
∂&ograve; ∂&ograve; ∂&ograve;
∂&igrave; ∂&icirc; ∂&iuml;
&iacute;&Uuml;&Eacute; &iacute;&ecirc;~&aring;&euml;&Ntilde;&ccedil;&ecirc;&atilde;~&iacute;&aacute;&ccedil;&aring;&euml; (&ntilde; I &oacute; I &ograve; ) → (&igrave;I &icirc; I &iuml; ) I ~&aring;&Ccedil; p &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc; &ccedil;&Ntilde; d &iuml;&Uuml;&aacute;&Aring;&Uuml; &Aring;~&aring; &Auml;&Eacute; &Aring;&ccedil;&atilde;&eacute;&igrave;&iacute;&Eacute;&Ccedil; &Auml;&oacute; &ntilde; = &ntilde; (&igrave;I &icirc; I &iuml; ) I
&oacute; = &oacute; (&igrave;I &icirc; I &iuml; )
&ograve; = &ograve; (&igrave;I &icirc; I &iuml; ) &aacute;&aring;&iacute;&ccedil; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; dK
1107. q&ecirc;&aacute;&eacute;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &aacute;&aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
q&Uuml;&Eacute; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; &Ntilde;&ccedil;&ecirc; &Aring;&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &aacute;&euml;
∂ (&ntilde; I &oacute; I &ograve; )
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; =
&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve; = &ecirc;&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve; K
∂(&ecirc;I θI &ograve; )
i&Eacute;&iacute; &iacute;&Uuml;&Eacute; &euml;&ccedil;&auml;&aacute;&Ccedil; d &aacute;&euml; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&Eacute;&Ccedil; ~&euml; &Ntilde;&ccedil;&auml;&auml;&ccedil;&iuml;&euml;W
(&ntilde; I &oacute; ) ∈ o I χ N (&ntilde; I &oacute; ) ≤ &ograve; ≤ χ O ( &ntilde; I &oacute; ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; o &aacute;&euml; &eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; d &ccedil;&aring;&iacute;&ccedil; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K q&Uuml;&Eacute;&aring;
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; = ∫∫∫ &Ntilde; (&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θI &ograve; )&ecirc;&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve;
d
p
χ O ( &ecirc; &Aring;&ccedil;&euml; θ I&ecirc; &euml;&aacute;&aring; θ )


(
)
&Ntilde;
&ecirc;
&Aring;&ccedil;&euml;
I
&ecirc;
&euml;&aacute;&aring;
I
&ograve;
&Ccedil;&ograve;
θ
θ
 &ecirc;&Ccedil;&ecirc;&Ccedil;θ K

∫∫  χ (&ecirc; &Aring;&ccedil;&euml;∫θI&ecirc; &euml;&aacute;&aring; θ )
o ( &ecirc; Iθ ) 
N

e&Eacute;&ecirc;&Eacute; p &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc; &ccedil;&Ntilde; d &aacute;&aring; &Aring;&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;K
=
1108. q&ecirc;&aacute;&eacute;&auml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &aacute;&aring; p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
q&Uuml;&Eacute; a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; &Ntilde;&ccedil;&ecirc; p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &aacute;&euml;
∂ (&ntilde; I &oacute; I &ograve; )
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; =
&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ = &ecirc; O &euml;&aacute;&aring; θ&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ
∂ (&ecirc;I θI ϕ)
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; =
d
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= ∫∫∫ &Ntilde; (&ecirc; &euml;&aacute;&aring; θ &Aring;&ccedil;&euml; ϕI &ecirc; &euml;&aacute;&aring; θ &euml;&aacute;&aring; ϕI &ecirc; &Aring;&ccedil;&euml; θ) &ecirc; O &euml;&aacute;&aring; θ&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ I
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &euml;&ccedil;&auml;&aacute;&Ccedil; p &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc; &ccedil;&Ntilde; d &aacute;&aring; &euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;K q&Uuml;&Eacute; ~&aring;&Ouml;&auml;&Eacute; θ &ecirc;~&aring;&Ouml;&Eacute;&euml; &Ntilde;&ecirc;&ccedil;&atilde; M &iacute;&ccedil; Oπ I &iacute;&Uuml;&Eacute; ~&aring;&Ouml;&auml;&Eacute; ϕ
&ecirc;~&aring;&Ouml;&Eacute;&euml; &Ntilde;&ecirc;&ccedil;&atilde; M &iacute;&ccedil; π K
Figure 202.
1109. s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ p&ccedil;&auml;&aacute;&Ccedil;
s = ∫∫∫ &Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;
d
1110. s&ccedil;&auml;&igrave;&atilde;&Eacute; &aacute;&aring; `&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
s = ∫∫∫ &ecirc;&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve;
p ( &ecirc; I θ I&ograve; )
1111. s&ccedil;&auml;&igrave;&atilde;&Eacute; &aacute;&aring; p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
s = ∫∫∫ &ecirc; O &euml;&aacute;&aring; θ&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ
p ( &ecirc; I θ Iϕ )
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1112. j~&euml;&euml; &ccedil;&Ntilde; ~ p&ccedil;&auml;&aacute;&Ccedil;
&atilde; = ∫∫∫ &micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;s I
d
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &euml;&ccedil;&auml;&aacute;&Ccedil; &ccedil;&Aring;&Aring;&igrave;&eacute;&aacute;&Eacute;&euml; ~ &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring; d ~&aring;&Ccedil; &aacute;&iacute;&euml; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; ~&iacute;
~ &eacute;&ccedil;&aacute;&aring;&iacute; (&ntilde; I &oacute; I &ograve; ) &aacute;&euml; &micro;(&ntilde; I &oacute; I &ograve; ) K
1113. `&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; j~&euml;&euml; &ccedil;&Ntilde; ~ p&ccedil;&auml;&aacute;&Ccedil;
j &oacute;&ograve;
j &ntilde;&oacute;
j
I &oacute; = &ntilde;&ograve; I &ograve; =
I
&ntilde;=
&atilde;
&atilde;
&atilde;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
j &oacute;&ograve; = ∫∫∫ &ntilde;&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I
d
j &ntilde;&ograve; = ∫∫∫ &oacute;&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I
d
j &ntilde;&oacute; = ∫∫∫ &ograve;&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s
d
~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&ecirc;&euml;&iacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; &eacute;&auml;~&aring;&Eacute;&euml; &ntilde; = M I
&oacute; = M I &ograve; = M I &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;I &micro;(&ntilde; I &oacute; I &ograve; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K
1114. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; f&aring;&Eacute;&ecirc;&iacute;&aacute;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute; E&ccedil;&ecirc; &ograve; = M FI &oacute;&ograve;-&eacute;&auml;~&aring;&Eacute;
E &ntilde; = M FI ~&aring;&Ccedil; &ntilde;&ograve;-&eacute;&auml;~&aring;&Eacute; E &oacute; = M F
f &ntilde;&oacute; = ∫∫∫ &ograve; O&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I
d
f &oacute;&ograve; = ∫∫∫ &ntilde; O&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I
d
f &ntilde;&ograve; = ∫∫∫ &oacute; O&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s K
d
1115. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; f&aring;&Eacute;&ecirc;&iacute;&aacute;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml;I &oacute;-~&ntilde;&aacute;&euml;I ~&aring;&Ccedil; &ograve;-~&ntilde;&aacute;&euml;
f &ntilde; = f &ntilde;&oacute; + f &ntilde;&ograve; = ∫∫∫ (&ograve; O + &oacute; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I
d
f &oacute; = f &ntilde;&oacute; + f &oacute;&ograve; = ∫∫∫ (&ograve; O + &ntilde; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I
d
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f&ograve; = f &ntilde;&ograve; + f &oacute;&ograve; = ∫∫∫ (&oacute; O + &ntilde; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s K
d
1116. m&ccedil;&auml;~&ecirc; j&ccedil;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; f&aring;&Eacute;&ecirc;&iacute;&aacute;~
fM = f &ntilde;&oacute; + f &oacute;&ograve; + f &ntilde;&ograve; = ∫∫∫ (&ntilde; O + &oacute; O + &ograve; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s
d
9.12 Line Integral
p&Aring;~&auml;~&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W c(&ntilde; I &oacute; I &ograve; ) I c(&ntilde; I &oacute; ) I &Ntilde; (&ntilde; )
p&Aring;~&auml;~&ecirc; &eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml;W &igrave;(&ntilde; I &oacute; I &ograve; )
`&igrave;&ecirc;&icirc;&Eacute;&euml;W `I `N I ` O
i&aacute;&atilde;&aacute;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;W ~I &Auml;I α I β
m~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;&euml;W &iacute;I &euml;
m&ccedil;&auml;~&ecirc; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W &ecirc; I θ
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W c (mI nI o )
r
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;W &ecirc; (&euml; )
r r r r
r&aring;&aacute;&iacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W &aacute; I &agrave; I &acirc; I τ
^&ecirc;&Eacute;~ &ccedil;&Ntilde; &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;W p
i&Eacute;&aring;&Ouml;&iacute;&Uuml; &ccedil;&Ntilde; ~ &Aring;&igrave;&ecirc;&icirc;&Eacute;W i
j~&euml;&euml; &ccedil;&Ntilde; ~ &iuml;&aacute;&ecirc;&Eacute;W &atilde;
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W ρ(&ntilde; I &oacute; I &ograve; ) I ρ(&ntilde; I &oacute; )
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &atilde;~&euml;&euml;W &ntilde; I &oacute; I &ograve;
c&aacute;&ecirc;&euml;&iacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W j &ntilde;&oacute; I j &oacute;&ograve; I j &ntilde;&ograve;
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W f &ntilde; I f &oacute; I f&ograve;
s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ &euml;&ccedil;&auml;&aacute;&Ccedil;W s
t&ccedil;&ecirc;&acirc;W t
r
j~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W _
`&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute;W f
b&auml;&Eacute;&Aring;&iacute;&ecirc;&ccedil;&atilde;&ccedil;&iacute;&aacute;&icirc;&Eacute; &Ntilde;&ccedil;&ecirc;&Aring;&Eacute;W ε
j~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring; &Ntilde;&auml;&igrave;&ntilde;W ψ
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1117. i&aacute;&aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ p&Aring;~&auml;~&ecirc; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
r r
i&Eacute;&iacute; ~ &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &Auml;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ecirc; = &ecirc; (&euml; ) I
M ≤ &euml; ≤ p I ~&aring;&Ccedil; ~ &euml;&Aring;~&auml;~&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; c &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; &ccedil;&icirc;&Eacute;&ecirc; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; `K
q&Uuml;&Eacute;&aring;
p
r
∫ c(&ecirc; (&euml; ))&Ccedil;&euml; = ∫ c(&ntilde; I &oacute;I &ograve; )&Ccedil;&euml; = ∫ c&Ccedil;&euml; I
M
`
`
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Ccedil;&euml; &aacute;&euml; &iacute;&Uuml;&Eacute; ~&ecirc;&Aring; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;K
∫ c &Ccedil;&euml; = ∫ c &Ccedil;&euml; + ∫ c &Ccedil;&euml;
1118.
`N ∪` O
`N
`O
Figure 203.
r r
1119. f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml; &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&euml; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil; &Auml;&oacute; &ecirc; = &ecirc; (&iacute; ) I
α ≤ &iacute; ≤ β I &iacute;&Uuml;&Eacute;&aring;
β
∫ c(&ntilde; I &oacute;I &ograve; )&Ccedil;&euml; = ∫ c(&ntilde; (&iacute; )I &oacute;(&iacute; )I &ograve;(&iacute; )) (&ntilde; ′(&iacute; )) + (&oacute;′(&iacute; )) + (&ograve;′(&iacute; )) &Ccedil;&iacute; K
O
O
O
α
`
1120. f&Ntilde; ` &aacute;&euml; ~ &euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml; &Aring;&igrave;&ecirc;&icirc;&Eacute; &aacute;&aring; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
&oacute; = &Ntilde; (&ntilde; ) I ~ ≤ &ntilde; ≤ &Auml; I &iacute;&Uuml;&Eacute;&aring;
&Auml;
∫ c(&ntilde; I &oacute; )&Ccedil;&euml; = ∫ c(&ntilde; I &Ntilde; (&ntilde; )) N + (&Ntilde; ′(&ntilde; )) &Ccedil;&ntilde; K
O
`
~
1121. i&aacute;&aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; p&Aring;~&auml;~&ecirc; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &aacute;&aring; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
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β
O
 &Ccedil;&ecirc; 
∫` c(&ntilde; I &oacute; )&Ccedil;&euml; = ∫α c(&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θ) &ecirc; +  &Ccedil;θ  &Ccedil;θ I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&auml;~&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ecirc;EθF K
O
1122. i&aacute;&aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; s&Eacute;&Aring;&iacute;&ccedil;&ecirc; c&aacute;&Eacute;&auml;&Ccedil;
r r
i&Eacute;&iacute; ~ &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &Auml;&Eacute; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ecirc; = &ecirc; (&euml; ) I
M ≤ &euml; ≤ p K q&Uuml;&Eacute;&aring;
r
&Ccedil;&ecirc; r
= τ = (&Aring;&ccedil;&euml; αI &Aring;&ccedil;&euml; βI &Aring;&ccedil;&euml; γ )
&Ccedil;&euml;
&aacute;&euml; &iacute;&Uuml;&Eacute; &igrave;&aring;&aacute;&iacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &iacute;~&aring;&Ouml;&Eacute;&aring;&iacute; &auml;&aacute;&aring;&Eacute; &iacute;&ccedil; &iacute;&Uuml;&aacute;&euml; &Aring;&igrave;&ecirc;&icirc;&Eacute;K
Figure 204.
r
i&Eacute;&iacute; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; c (mI nI o ) &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; &ccedil;&icirc;&Eacute;&ecirc; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; `K
r
q&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; c ~&auml;&ccedil;&aring;&Ouml; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute;
` &aacute;&euml;
p
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve; = ∫ (m &Aring;&ccedil;&euml; α + n &Aring;&ccedil;&euml; β + o &Aring;&ccedil;&euml; γ )&Ccedil;&euml; K
`
M
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CHAPTER 9. INTEGRAL CALCULUS
1123. m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml; &ccedil;&Ntilde; i&aacute;&aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml; &ccedil;&Ntilde; s&Eacute;&Aring;&iacute;&ccedil;&ecirc; c&aacute;&Eacute;&auml;&Ccedil;&euml;
r r
r r
c
⋅
&Ccedil;
&ecirc;
=
−
c
∫
∫ ⋅ &Ccedil;&ecirc; I
(
−`
)
(
)
`
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; -` &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; &iuml;&aacute;&iacute;&Uuml; &iacute;&Uuml;&Eacute; &ccedil;&eacute;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute; &ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;K
r
r
r
r
∫ (c ⋅ &Ccedil;&ecirc; ) = ∫ (c ⋅ &Ccedil;&ecirc; ) = ∫ (c ⋅ &Ccedil;&ecirc; ) + ∫ (c ⋅ &Ccedil;&ecirc; ) I
r
r
`N ∪ ` O
`
r
`N
r
`O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ` &aacute;&euml; &iacute;&Uuml;&Eacute; &igrave;&aring;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute;&euml; `N ~&aring;&Ccedil; ` O K
r
1124. f&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&euml; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil; &Auml;&oacute; &ecirc; (&iacute; ) = &ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ) I
α ≤ &iacute; ≤ β I &iacute;&Uuml;&Eacute;&aring;
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve; =
`
β
&Ccedil;&oacute;
&Ccedil;&ntilde;
&Ccedil;&ograve; 

= ∫  m(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + n(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + o(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) &Ccedil;&iacute;
&Ccedil;&iacute;
&Ccedil;&iacute;
&Ccedil;&iacute; 
α
1125. f&Ntilde; ` &auml;&aacute;&Eacute;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute; ~&aring;&Ccedil; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &oacute; = &Ntilde; (&ntilde; ) I
&iacute;&Uuml;&Eacute;&aring;
&Auml;
&Ccedil;&Ntilde; 

m&Ccedil;&ntilde;
n&Ccedil;&oacute;
+
=
∫`
∫~  m(&ntilde; I &Ntilde; (&ntilde; )) + n(&ntilde; I &Ntilde; (&ntilde; )) &Ccedil;&ntilde; &Ccedil;&ntilde; K
1126. d&ecirc;&Eacute;&Eacute;&aring;∞&euml; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;
 ∂n ∂m 
∫∫o  ∂&ntilde; − ∂&oacute; &Ccedil;&ntilde;&Ccedil;&oacute; = ∫` m&Ccedil;&ntilde; + n&Ccedil;&oacute; I
r
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; c = m(&ntilde; I &oacute; ) &aacute; + n(&ntilde; I &oacute; ) &agrave; &aacute;&euml; ~ &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;∂m ∂n
&iacute;&aacute;&ccedil;&aring; &iuml;&aacute;&iacute;&Uuml; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &Ntilde;&aacute;&ecirc;&euml;&iacute; &eacute;~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;
I
&aacute;&aring; ~
∂&oacute; ∂&ntilde;
&euml;&ccedil;&atilde;&Eacute; &Ccedil;&ccedil;&atilde;~&aacute;&aring; oI &iuml;&Uuml;&aacute;&Aring;&Uuml; &aacute;&euml; &Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; ~ &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;I &eacute;&aacute;&Eacute;&Aring;&Eacute;&iuml;&aacute;&euml;&Eacute;
&euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml; &Aring;&igrave;&ecirc;&icirc;&Eacute; `K
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CHAPTER 9. INTEGRAL CALCULUS
1127. ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ o&Eacute;&Ouml;&aacute;&ccedil;&aring; o _&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; `&igrave;&ecirc;&icirc;&Eacute; `
N
p = ∫∫ &Ccedil;&ntilde;&Ccedil;&oacute; = ∫ &ntilde;&Ccedil;&oacute; − &oacute;&Ccedil;&ntilde;
O`
o
1128. m~&iacute;&Uuml; f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; i&aacute;&aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;
r
r
r
r
q&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; c = m &aacute; + n &agrave; + o&acirc; &aacute;&euml;
&euml;~&aacute;&Ccedil; &iacute;&ccedil; &Auml;&Eacute; &eacute;~&iacute;&Uuml; &aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;I &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde; mI nI ~&aring;&Ccedil; o ~&ecirc;&Eacute;
&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &aacute;&aring; ~ &Ccedil;&ccedil;&atilde;~&aacute;&aring; aI ~&aring;&Ccedil; &aacute;&Ntilde; &iacute;&Uuml;&Eacute;&ecirc;&Eacute; &Eacute;&ntilde;&aacute;&euml;&iacute;&euml; &euml;&ccedil;&atilde;&Eacute; &euml;&Aring;~&auml;~&ecirc;
&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &igrave; = &igrave;(&ntilde; I &oacute; I &ograve; ) E~ &euml;&Aring;~&auml;~&ecirc; &eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml;F &aacute;&aring; a &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute;
r
∂&igrave;
∂&igrave;
∂&igrave;
=nI
c = &Ouml;&ecirc;~&Ccedil; &igrave; I &ccedil;&ecirc;
=oK
=mI
∂&ograve;
∂&oacute;
∂&ntilde;
q&Uuml;&Eacute;&aring;
rr r
∫ c(&ecirc; ) ⋅ &Ccedil;&ecirc; = ∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve; = &igrave;(_) − &igrave;(^ ) K
`
`
1129. q&Eacute;&euml;&iacute; &Ntilde;&ccedil;&ecirc; ~ `&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute; c&aacute;&Eacute;&auml;&Ccedil;
r
^ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde; c = &Ouml;&ecirc;~&Ccedil; &igrave; &aacute;&euml; &Aring;~&auml;&auml;&Eacute;&Ccedil; ~ &Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute;
r
r
r
r
&Ntilde;&aacute;&Eacute;&auml;&Ccedil;K q&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; c = m &aacute; + n &agrave; + o&acirc;
&aacute;&euml; &eacute;~&iacute;&Uuml; &aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &aacute;&Ntilde; ~&aring;&Ccedil; &ccedil;&aring;&auml;&oacute; &aacute;&Ntilde;
r
r
r
&aacute;
&agrave;
&acirc;
r ∂
∂
∂ r
&Aring;&igrave;&ecirc;&auml; c =
=MK
∂&ntilde; ∂&oacute; ∂&ograve;
m n o
f&Ntilde; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &aacute;&euml; &iacute;~&acirc;&Eacute;&aring; &aacute;&aring; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute; &euml;&ccedil; &iacute;&Uuml;~&iacute;
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; = &igrave;(_) − &igrave;(^ ) I
`
&iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &iacute;&Eacute;&euml;&iacute; &Ntilde;&ccedil;&ecirc; &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&aacute;&aring;&Ouml; &aacute;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; &aacute;&euml; &Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute;
&Aring;~&aring; &Auml;&Eacute; &iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring; &aacute;&aring; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde;
∂m ∂n
=
K
∂&oacute; ∂&ntilde;
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CHAPTER 9. INTEGRAL CALCULUS
1130. i&Eacute;&aring;&Ouml;&iacute;&Uuml; &ccedil;&Ntilde; ~ `&igrave;&ecirc;&icirc;&Eacute;
r
O
O
O
β
β
&Ccedil;&oacute;
&Ccedil;&ecirc;
i = ∫ &Ccedil;&euml; = ∫
(&iacute; ) &Ccedil;&iacute; = ∫  &Ccedil;&ntilde;  +   +  &Ccedil;&ograve;  &Ccedil;&iacute; I
&Ccedil;&iacute;
&Ccedil;&iacute;   &Ccedil;&iacute;   &Ccedil;&iacute; 
`
α
α 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ` &aacute;~ ~ &eacute;&aacute;&Eacute;&Aring;&Eacute;&iuml;&aacute;&euml;&Eacute; &euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml; &Aring;&igrave;&ecirc;&icirc;&Eacute; &Ccedil;&Eacute;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&euml;&aacute;r
&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ecirc; (&iacute; ) I α ≤ &iacute; ≤ β K
f&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&euml; &iacute;&iuml;&ccedil;-&Ccedil;&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml;I &iacute;&Uuml;&Eacute;&aring;
r
O
O
β
β
&Ccedil;&ecirc;
 &Ccedil;&ntilde;   &Ccedil;&oacute; 
(&iacute; ) &Ccedil;&iacute; = ∫   +   &Ccedil;&iacute; K
i = ∫ &Ccedil;&euml; = ∫
&Ccedil;&iacute;
&Ccedil;&iacute;   &Ccedil;&iacute; 
α
α 
`
f&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&euml; &iacute;&Uuml;&Eacute; &Ouml;&ecirc;~&eacute;&Uuml; &ccedil;&Ntilde; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &oacute; = &Ntilde; (&ntilde; ) &aacute;&aring; &iacute;&Uuml;&Eacute; &ntilde;&oacute;&eacute;&auml;~&aring;&Eacute; (~ ≤ &ntilde; ≤ &Auml;) I &iacute;&Uuml;&Eacute;&aring;
&Auml;
i=∫
~
O
 &Ccedil;&oacute; 
N +   &Ccedil;&ntilde; K
 &Ccedil;&ntilde; 
1131. i&Eacute;&aring;&Ouml;&iacute;&Uuml; &ccedil;&Ntilde; ~ `&igrave;&ecirc;&icirc;&Eacute; &aacute;&aring; m&ccedil;&auml;~&ecirc; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;
β
O
 &Ccedil;&ecirc; 
i = ∫   + &ecirc; O &Ccedil;θ I
&Ccedil;θ 
α 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ecirc; = &ecirc;(θ) I
α ≤ θ ≤ β &aacute;&aring; &eacute;&ccedil;&auml;~&ecirc; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;K
1132. j~&euml;&euml; &ccedil;&Ntilde; ~ t&aacute;&ecirc;&Eacute;
&atilde; = ∫ ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I
`
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; ρ(&ntilde; I &oacute; I &ograve; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &atilde;~&euml;&euml; &eacute;&Eacute;&ecirc; &igrave;&aring;&aacute;&iacute; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &iuml;&aacute;&ecirc;&Eacute;K
f&Ntilde; ` &aacute;&euml; ~ &Aring;&igrave;&ecirc;&icirc;&Eacute; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
r
&ecirc; (&iacute; ) = &ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ) I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &atilde;~&euml;&euml; &Aring;~&aring; &Auml;&Eacute; &Aring;&ccedil;&atilde;&eacute;&igrave;&iacute;&Eacute;&Ccedil; &Auml;&oacute;
&iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~
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CHAPTER 9. INTEGRAL CALCULUS
β
&atilde; = ∫ ρ(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ))
α
O
O
O
 &Ccedil;&ntilde;   &Ccedil;&oacute;   &Ccedil;&ograve; 
  +   +   &Ccedil;&iacute; K
 &Ccedil;&iacute;   &Ccedil;&iacute;   &Ccedil;&iacute; 
f&Ntilde; ` &aacute;&euml; ~ &Aring;&igrave;&ecirc;&icirc;&Eacute; &aacute;&aring; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &atilde;~&euml;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &iuml;&aacute;&ecirc;&Eacute; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring;
&Auml;&oacute;
&atilde; = ∫ ρ(&ntilde; I &oacute; )&Ccedil;&euml; I
`
&ccedil;&ecirc;
β
&atilde; = ∫ ρ(&ntilde; (&iacute; )I &oacute; (&iacute; ))
α
O
O
 &Ccedil;&ntilde;   &Ccedil;&oacute; 
  +   &Ccedil;&iacute; E&aacute;&aring; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&ccedil;&ecirc;&atilde;FK
 &Ccedil;&iacute;   &Ccedil;&iacute; 
1133. `&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; j~&euml;&euml; &ccedil;&Ntilde; ~ t&aacute;&ecirc;&Eacute;
j &oacute;&ograve;
j &ntilde;&oacute;
j
I &oacute; = &ntilde;&ograve; I &ograve; =
I
&ntilde;=
&atilde;
&atilde;
&atilde;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
j &oacute;&ograve; = ∫ &ntilde;ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I
`
j &ntilde;&ograve; = ∫ &oacute;ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I
`
j &ntilde;&oacute; = ∫ &ograve;ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; K
`
1134. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; f&aring;&Eacute;&ecirc;&iacute;&aacute;~
q&Uuml;&Eacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml;I &oacute;-~&ntilde;&aacute;&euml;I ~&aring;&Ccedil; &ograve;-~&ntilde;&aacute;&euml;
~&ecirc;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;
f &ntilde; = ∫ (&oacute; O + &ograve; O )ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I
`
f &oacute; = ∫ (&ntilde; O + &ograve; O )ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I
`
f &ograve; = ∫ (&ntilde; O + &oacute; O )ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; K
`
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CHAPTER 9. INTEGRAL CALCULUS
1135. ^&ecirc;&Eacute;~ &ccedil;&Ntilde; ~ o&Eacute;&Ouml;&aacute;&ccedil;&aring; _&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; ~ `&auml;&ccedil;&euml;&Eacute;&Ccedil; `&igrave;&ecirc;&icirc;&Eacute;
N
p = ∫ &ntilde;&Ccedil;&oacute; = − ∫ &oacute;&Ccedil;&ntilde; = ∫ &ntilde;&Ccedil;&oacute; − &oacute;&Ccedil;&ntilde; K
O`
`
`
Figure 205.
f&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil; &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &aacute;&aring; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&ccedil;&ecirc;&atilde;
r
&ecirc; (&iacute; ) = &ntilde; (&iacute; )I &oacute; (&iacute; ) I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; ~&ecirc;&Eacute;~ &Aring;~&aring; &Auml;&Eacute; &Aring;~&auml;&Aring;&igrave;&auml;~&iacute;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~
β
β
β
&Ccedil;&oacute;
&Ccedil;&oacute;
&Ccedil;&ntilde;
N 
&Ccedil;&ntilde; 
p = ∫ &ntilde; (&iacute; ) &Ccedil;&iacute; = − ∫ &oacute; (&iacute; ) &Ccedil;&iacute; = ∫  &ntilde; (&iacute; ) − &oacute; (&iacute; ) &Ccedil;&iacute; K
&Ccedil;&iacute;
&Ccedil;&iacute;
O α
&Ccedil;&iacute;
&Ccedil;&iacute; 
α
α
1136. s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ p&ccedil;&auml;&aacute;&Ccedil; c&ccedil;&ecirc;&atilde;&Eacute;&Ccedil; &Auml;&oacute; o&ccedil;&iacute;~&iacute;&aacute;&aring;&Ouml; ~ `&auml;&ccedil;&euml;&Eacute;&Ccedil; `&igrave;&ecirc;&icirc;&Eacute;
~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml;
π
s = − π ∫ &oacute; O&Ccedil;&ntilde; = −Oπ ∫ &ntilde;&oacute;&Ccedil;&oacute; = − ∫ O&ntilde;&oacute;&Ccedil;&oacute; + &oacute; O&Ccedil;&ntilde;
O`
`
`
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Figure 206.
1137. t&ccedil;&ecirc;&acirc;
r
t&ccedil;&ecirc;&acirc; &Ccedil;&ccedil;&aring;&Eacute; &Auml;&oacute; ~ &Ntilde;&ccedil;&ecirc;&Aring;&Eacute; c &ccedil;&aring; ~&aring; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute; &atilde;&ccedil;&icirc;&aacute;&aring;&Ouml; ~&auml;&ccedil;&aring;&Ouml; ~ &Aring;&igrave;&ecirc;&icirc;&Eacute;
` &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;
r r
t = ∫ c ⋅ &Ccedil;&ecirc; I
`
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; c &aacute;&euml; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&ccedil;&ecirc;&Aring;&Eacute; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; ~&Aring;&iacute;&aacute;&aring;&Ouml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;I &Ccedil; &ecirc; &aacute;&euml;
&iacute;&Uuml;&Eacute; &igrave;&aring;&aacute;&iacute; &iacute;~&aring;&Ouml;&Eacute;&aring;&iacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;K
Figure 207.
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CHAPTER 9. INTEGRAL CALCULUS
f&Ntilde; &iacute;&Uuml;&Eacute; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute; &aacute;&euml; &atilde;&ccedil;&icirc;&Eacute;&Ccedil; ~&auml;&ccedil;&aring;&Ouml; ~ &Aring;&igrave;&ecirc;&icirc;&Eacute; ` &aacute;&aring; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;I &iacute;&Uuml;&Eacute;&aring;
r r
t = ∫ c ⋅ &Ccedil; &ecirc; = ∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; I
`
`
f&Ntilde; ~ &eacute;~&iacute;&Uuml; ` &aacute;&euml; &euml;&eacute;&Eacute;&Aring;&aacute;&Ntilde;&aacute;&Eacute;&Ccedil; &Auml;&oacute; ~ &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc; &iacute; E&iacute; &ccedil;&Ntilde;&iacute;&Eacute;&aring; &atilde;&Eacute;~&aring;&euml;
&iacute;&aacute;&atilde;&Eacute;FI &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~ &Ntilde;&ccedil;&ecirc; &Aring;~&auml;&Aring;&igrave;&auml;~&iacute;&aacute;&aring;&Ouml; &iuml;&ccedil;&ecirc;&acirc; &Auml;&Eacute;&Aring;&ccedil;&atilde;&Eacute;&euml;
β
&Ccedil;&ntilde;
&Ccedil;&oacute;
&Ccedil;&ograve; 

t = ∫ m(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + n(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + o(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ))  &Ccedil;&iacute;I
&Ccedil;&iacute;
&Ccedil;&iacute;
&Ccedil;&iacute; 
α
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute; &Ouml;&ccedil;&Eacute;&euml; &Ntilde;&ecirc;&ccedil;&atilde; α &iacute;&ccedil; β K
r
f&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; c &aacute;&euml; &Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute; ~&aring;&Ccedil; &igrave;(&ntilde; I &oacute; I &ograve; ) &aacute;&euml; ~ &euml;&Aring;~&auml;~&ecirc;
&eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &iuml;&ccedil;&ecirc;&acirc; &ccedil;&aring; ~&aring; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute; &atilde;&ccedil;&icirc;&aacute;&aring;&Ouml;
&Ntilde;&ecirc;&ccedil;&atilde; ^ &iacute;&ccedil; _ &Aring;~&aring; &Auml;&Eacute; &Ntilde;&ccedil;&igrave;&aring;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~
t = &igrave;(_ ) − &igrave;(^ ) K
1138. ^&atilde;&eacute;&Eacute;&ecirc;&Eacute;∞&euml; i~&iuml;
r r
_
∫ ⋅ &Ccedil;&ecirc; = &micro;Mf K
`
r
q&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ &atilde;~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; _ ~&ecirc;&ccedil;&igrave;&aring;&Ccedil; ~ &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil; &eacute;~&iacute;&Uuml;
` &aacute;&euml; &Eacute;&egrave;&igrave;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &iacute;&ccedil;&iacute;~&auml; &Aring;&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute; f &Ntilde;&auml;&ccedil;&iuml;&aacute;&aring;&Ouml; &iacute;&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml; &iacute;&Uuml;&Eacute; ~&ecirc;&Eacute;~
&Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &eacute;~&iacute;&Uuml;K
Figure 208.
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CHAPTER 9. INTEGRAL CALCULUS
1139. c~&ecirc;~&Ccedil;~&oacute;∞&euml; i~&iuml;
r r
&Ccedil;ψ
ε = ∫ b ⋅ &Ccedil;&ecirc; = −
&Ccedil;&iacute;
`
q&Uuml;&Eacute; &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&ccedil;&atilde;&ccedil;&iacute;&aacute;&icirc;&Eacute; &Ntilde;&ccedil;&ecirc;&Aring;&Eacute; E&Eacute;&atilde;&Ntilde;F ε &aacute;&aring;&Ccedil;&igrave;&Aring;&Eacute;&Ccedil; ~&ecirc;&ccedil;&igrave;&aring;&Ccedil; ~ &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;
&auml;&ccedil;&ccedil;&eacute; ` &aacute;&euml; &Eacute;&egrave;&igrave;~&auml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &ecirc;~&iacute;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&Uuml;~&aring;&Ouml;&Eacute; &ccedil;&Ntilde; &atilde;~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring; &Ntilde;&auml;&igrave;&ntilde; ψ
&eacute;~&euml;&euml;&aacute;&aring;&Ouml; &iacute;&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml; &iacute;&Uuml;&Eacute; &auml;&ccedil;&ccedil;&eacute;K
Figure 209.
9.13 Surface Integral
p&Aring;~&auml;~&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde; (&ntilde; I &oacute; I &ograve; ) I &ograve; (&ntilde; I &oacute; )
r
r
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W &ecirc; (&igrave;I &icirc; ) I &ecirc; (&ntilde; I &oacute; I &ograve; )
r r r
r&aring;&aacute;&iacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W &aacute; I &agrave; I &acirc;
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W p
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W c (mI nI o )
r
r
a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W &Ccedil;&aacute;&icirc; c = ∇ ⋅ c
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r
r
`&igrave;&ecirc;&auml; &ccedil;&Ntilde; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W &Aring;&igrave;&ecirc;&auml; c = ∇ &times; c
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; ~ &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W &Ccedil;p
r
k&ccedil;&ecirc;&atilde;~&auml; &iacute;&ccedil; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W &aring;
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~W ^
j~&euml;&euml; &ccedil;&Ntilde; ~ &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W &atilde;
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W &micro;(&ntilde; I &oacute; I &ograve; )
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml; &ccedil;&Ntilde; &Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; &atilde;~&euml;&euml;W &ntilde; I &oacute; I &ograve;
c&aacute;&ecirc;&euml;&iacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W j &ntilde;&oacute; I j &oacute;&ograve; I j &ntilde;&ograve;
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; &aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W f &ntilde;&oacute; I f &oacute;&ograve; I f &ntilde;&ograve; I f &ntilde; I f &oacute; I f&ograve;
s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ &euml;&ccedil;&auml;&aacute;&Ccedil;W s
r
c&ccedil;&ecirc;&Aring;&Eacute;W c
d&ecirc;~&icirc;&aacute;&iacute;~&iacute;&aacute;&ccedil;&aring;~&auml; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;W d
rr
c&auml;&igrave;&aacute;&Ccedil; &icirc;&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;W &icirc; (&ecirc; )
c&auml;&igrave;&aacute;&Ccedil; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W ρ
r
m&ecirc;&Eacute;&euml;&euml;&igrave;&ecirc;&Eacute;W &eacute;(&ecirc; )
j~&euml;&euml; &Ntilde;&auml;&igrave;&ntilde;I &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&auml;&igrave;&ntilde;W Φ
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &Aring;&Uuml;~&ecirc;&Ouml;&Eacute;W n
`&Uuml;~&ecirc;&Ouml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W σ(&ntilde; I &oacute; )
r
j~&Ouml;&aring;&aacute;&iacute;&igrave;&Ccedil;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;W b
1140. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ p&Aring;~&auml;~&ecirc; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
i&Eacute;&iacute; ~ &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &Auml;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;
r
r
r
r
&ecirc; (&igrave;I &icirc; ) = &ntilde; (&igrave;I &icirc; )&aacute; + &oacute; (&igrave;I &icirc; ) &agrave; + &ograve; (&igrave;I &icirc; )&acirc; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; (&igrave;I &icirc; ) &ecirc;~&aring;&Ouml;&Eacute;&euml; &ccedil;&icirc;&Eacute;&ecirc; &euml;&ccedil;&atilde;&Eacute; &Ccedil;&ccedil;&atilde;~&aacute;&aring; a(&igrave;I &icirc; ) &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &igrave;&icirc;&eacute;&auml;~&aring;&Eacute;K
q&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; ~ &euml;&Aring;~&auml;~&ecirc; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &Ntilde; (&ntilde; I &oacute; I &ograve; ) &ccedil;&icirc;&Eacute;&ecirc;
&iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &aacute;&euml; &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil; ~&euml;
r
r
∂&ecirc; ∂&ecirc;
(
)
(
(
)
(
)
(
)
)
=
&times;
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;p
&Ntilde;
&ntilde;
&igrave;
I
&icirc;
I
&oacute;
&igrave;
I
&icirc;
I
&ograve;
&igrave;
I
&icirc;
&Ccedil;&igrave;&Ccedil;&icirc; I
∫∫p
∫∫
∂
∂
&igrave;
&icirc;
a(&igrave; I &icirc; )
r
r
∂&ecirc;
∂&ecirc;
~&aring;&Ccedil;
~&ecirc;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;
∂&igrave;
∂&icirc;
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r
r
r ∂&oacute;
r ∂&ograve;
∂&ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) &aacute; + (&igrave;I &icirc; ) &agrave; + (&igrave;I &icirc; )&acirc; I
∂&igrave;
∂&igrave; ∂&igrave;
∂&igrave;
r
r
r ∂&oacute;
r ∂&ograve;
∂ &ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) &aacute; + (&igrave;I &icirc; ) &agrave; + (&igrave;I &icirc; )&acirc;
∂&icirc;
∂&icirc;
∂&icirc; ∂&icirc;
r
r
∂&ecirc; ∂&ecirc;
~&aring;&Ccedil;
&aacute;&euml; &iacute;&Uuml;&Eacute; &Aring;&ecirc;&ccedil;&euml;&euml; &eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;K
&times;
∂&igrave; ∂&icirc;
1141. f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ograve; = &ograve;(&ntilde; I &oacute; ) &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&ograve; (&ntilde; I &oacute; ) &aacute;&euml; ~ &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &aacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&ccedil;&atilde;~&aacute;&aring; a(&ntilde; I &oacute; ) I
&iacute;&Uuml;&Eacute;&aring;
∫∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;p = (∫∫ ) &Ntilde; (&ntilde; I &oacute; I &ograve;(&ntilde; I &oacute; ))
p
a &ntilde; I&oacute;
O
 ∂&ograve;   ∂&ograve; 
N +   +   &Ccedil;&ntilde;&Ccedil;&oacute; K
 ∂&ntilde;   ∂&oacute; 
O
r
1142. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; s&Eacute;&Aring;&iacute;&ccedil;&ecirc; c&aacute;&Eacute;&auml;&Ccedil; c &ccedil;&icirc;&Eacute;&ecirc; &iacute;&Uuml;&Eacute; p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p
• f&Ntilde; p &aacute;&euml; &ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil; &ccedil;&igrave;&iacute;&iuml;~&ecirc;&Ccedil;I &iacute;&Uuml;&Eacute;&aring;
r
r
r
r
(
)
c
&ntilde;
I
&oacute;
I
&ograve;
⋅
&Ccedil;
p
=
c
∫∫
∫∫ (&ntilde; I &oacute;I &ograve; ) ⋅ &aring;&Ccedil;p
p
•
p
r
r
r
 ∂&ecirc; ∂&ecirc; 
= ∫∫ c(&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )I &ograve; (&igrave;I &icirc; )) ⋅  &times;  &Ccedil;&igrave;&Ccedil;&icirc; K
 ∂&igrave; ∂&icirc; 
a( &igrave; I &icirc; )
f&Ntilde; p &aacute;&euml; &ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil; &aacute;&aring;&iuml;~&ecirc;&Ccedil;I &iacute;&Uuml;&Eacute;&aring;
r
r
r
r
∫∫ c(&ntilde; I &oacute;I &ograve; ) ⋅ &Ccedil;p = ∫∫ c(&ntilde; I &oacute;I &ograve; ) ⋅ &aring;&Ccedil;p
p
p
r
r
r
 ∂&ecirc; ∂&ecirc; 
= ∫∫ c(&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )I &ograve; (&igrave;I &icirc; )) ⋅  &times;  &Ccedil;&igrave;&Ccedil;&icirc; K
 ∂&icirc; ∂&igrave; 
a( &igrave; I &icirc; )
r r
&Ccedil;p = &aring;&Ccedil;p &aacute;&euml; &Aring;~&auml;&auml;&Eacute;&Ccedil; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;K a&ccedil;&iacute;
&atilde;&Eacute;~&aring;&euml; &iacute;&Uuml;&Eacute; &euml;&Aring;~&auml;~&ecirc; &eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; ~&eacute;&eacute;&ecirc;&ccedil;&eacute;&ecirc;&aacute;~&iacute;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;K
r
r
∂&ecirc;
∂&ecirc;
~&aring;&Ccedil;
~&ecirc;&Eacute; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
q&Uuml;&Eacute; &eacute;~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;
∂&igrave;
∂&icirc;
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r
r
r ∂&oacute;
r ∂&ograve;
∂&ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) ⋅ &aacute; + (&igrave;I &icirc; ) ⋅ &agrave; + (&igrave;I &icirc; ) ⋅ &acirc; I
∂&igrave; ∂&igrave;
∂&igrave;
∂&igrave;
r
r
r ∂&oacute;
r ∂&ograve;
∂ &ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) ⋅ &aacute; + (&igrave;I &icirc; ) ⋅ &agrave; + (&igrave;I &icirc; ) ⋅ &acirc; K
∂&icirc;
∂&icirc;
∂&icirc; ∂&icirc;
1143. f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &ograve; = &ograve; (&ntilde; I &oacute; ) I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&ograve; (&ntilde; I &oacute; ) &aacute;&euml; ~ &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &aacute;&aring; &iacute;&Uuml;&Eacute; &Ccedil;&ccedil;&atilde;~&aacute;&aring; a(&ntilde; I &oacute; ) I
&iacute;&Uuml;&Eacute;&aring;
• f&Ntilde; p &aacute;&euml; &ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil; &igrave;&eacute;&iuml;~&ecirc;&Ccedil;I &aacute;K&Eacute;K &iacute;&Uuml;&Eacute; &acirc;-&iacute;&Uuml; &Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute;
&aring;&ccedil;&ecirc;&atilde;~&auml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &aacute;&euml; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;I &iacute;&Uuml;&Eacute;&aring;
r
r
r
r
(
)
c
&ntilde;
I
&oacute;
I
&ograve;
⋅
&Ccedil;
p
=
c
∫∫
∫∫ (&ntilde; I &oacute;I &ograve; ) ⋅ &aring;&Ccedil;p
p
p
=
r
 ∂&ograve; r ∂&ograve; r r 
 −
(
)
c
&ntilde;
I
&oacute;
I
&ograve;
&aacute;−
&agrave; + &acirc; &Ccedil;&ntilde;&Ccedil;&oacute; I
⋅
∫∫
∂&oacute;
 ∂&ntilde;

a( &ntilde; I &oacute; )
f&Ntilde; p &aacute;&euml; &ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil; &Ccedil;&ccedil;&iuml;&aring;&iuml;~&ecirc;&Ccedil;I &aacute;K&Eacute;K &iacute;&Uuml;&Eacute; &acirc;-&iacute;&Uuml; &Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute;
&aring;&ccedil;&ecirc;&atilde;~&auml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &aacute;&euml; &aring;&Eacute;&Ouml;~&iacute;&aacute;&icirc;&Eacute;I &iacute;&Uuml;&Eacute;&aring;
r
r
r
r
∫∫ c(&ntilde; I &oacute;I &ograve; ) ⋅ &Ccedil;p = ∫∫ c(&ntilde; I &oacute;I &ograve; ) ⋅ &aring;&Ccedil;p
•
p
p
r
 ∂&ograve; r ∂&ograve; r r 
&agrave; − &acirc; &Ccedil;&ntilde;&Ccedil;&oacute; K
= ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅  &aacute; +
∂&oacute;
 ∂&ntilde;

a( &ntilde; I &oacute; )
1144.
∫∫ (c ⋅ &aring;)&Ccedil;p = ∫∫ m&Ccedil;&oacute;&Ccedil;&ograve; + n&Ccedil;&ograve;&Ccedil;&ntilde; + o&Ccedil;&ntilde;&Ccedil;&oacute;
r r
p
p
= ∫∫ (m &Aring;&ccedil;&euml; α + n &Aring;&ccedil;&euml; β + o &Aring;&ccedil;&euml; γ )&Ccedil;p I
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; m(&ntilde; I &oacute; I &ograve; ) I n(&ntilde; I &oacute; I &ograve; ) I o(&ntilde; I &oacute; I &ograve; ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde;
r
&iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; c K
&Aring;&ccedil;&euml; α I &Aring;&ccedil;&euml; β I &Aring;&ccedil;&euml; γ ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; ~&aring;&Ouml;&auml;&Eacute;&euml; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &iacute;&Uuml;&Eacute; &ccedil;&igrave;&iacute;&Eacute;&ecirc; &igrave;&aring;&aacute;&iacute;
r
&aring;&ccedil;&ecirc;&atilde;~&auml; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &aring; ~&aring;&Ccedil; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml;I &oacute;-~&ntilde;&aacute;&euml;I ~&aring;&Ccedil; &ograve;-~&ntilde;&aacute;&euml;I &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;K
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CHAPTER 9. INTEGRAL CALCULUS
1145. f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &aacute;&aring; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&ccedil;&ecirc;&atilde; &Auml;&oacute; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;
r
&ecirc; (&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )I &ograve; (&igrave;I &icirc; )) I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &auml;~&iacute;&iacute;&Eacute;&ecirc; &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~ &Aring;~&aring; &Auml;&Eacute;
&iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring; ~&euml;
m n
∂&ntilde; ∂&oacute;
∫∫p
∫∫ ∂&igrave; ∂&igrave;
p
a(&igrave; I &icirc; )
∂&ntilde; ∂&oacute;
∂&icirc; ∂&icirc;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; (&igrave;I &icirc; ) &ecirc;~&aring;&Ouml;&Eacute;&euml; &ccedil;&icirc;&Eacute;&ecirc; &euml;&ccedil;&atilde;&Eacute; &Ccedil;&ccedil;&atilde;~&aacute;&aring; a(&igrave;I &icirc; ) &ccedil;&Ntilde; &iacute;&Uuml;&Eacute;
&eacute;&auml;~&aring;&Eacute;K
( )
r r
c ⋅ &aring; &Ccedil;p = ∫∫ m&Ccedil;&oacute;&Ccedil;&ograve; + n&Ccedil;&ograve;&Ccedil;&ntilde; + o&Ccedil;&ntilde;&Ccedil;&oacute; =
o
∂&ograve;
&Ccedil;&igrave;&Ccedil;&icirc; I
∂&igrave;
∂&ograve;
∂&icirc;
&igrave;&icirc;-
1146. a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;
r r
r
c
⋅
&Ccedil;
p
=
∇
⋅
c
&Ccedil;s I
∫∫
∫∫∫
(
p
)
d
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
r
c(&ntilde; I &oacute; I &ograve; ) = m(&ntilde; I &oacute; I &ograve; )I n(&ntilde; I &oacute; I &ograve; )I o(&ntilde; I &oacute; I &ograve; )
&aacute;&euml; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; &iuml;&Uuml;&ccedil;&euml;&Eacute; &Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&euml; mI nI ~&aring;&Ccedil; o &Uuml;~&icirc;&Eacute;
&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &eacute;~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;I
r ∂m ∂n ∂o
∇⋅c =
+
+
∂&ntilde; ∂&oacute; ∂&ograve;
r
r
&aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; c I ~&auml;&euml;&ccedil; &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil; &Ccedil;&aacute;&icirc;c K q&Uuml;&Eacute; &euml;&oacute;&atilde;&Auml;&ccedil;&auml;
∫∫ &aacute;&aring;&Ccedil;&aacute;&Aring;~&iacute;&Eacute;&euml; &iacute;&Uuml;~&iacute; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &aacute;&euml; &iacute;~&acirc;&Eacute;&aring; &ccedil;&icirc;&Eacute;&ecirc; ~ &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;
&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;K
1147. a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde; &aacute;&aring; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; c&ccedil;&ecirc;&atilde;
 ∂m ∂n ∂o 

&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; K
+
+
=
+
m&Ccedil;&oacute;&Ccedil;&ograve;
n&Ccedil;&ntilde;&Ccedil;&ograve;
o&Ccedil;&ntilde;&Ccedil;&oacute;
+
∫∫p
∫∫∫
∂
∂
&ntilde;
&oacute;
&ograve;
∂

d 
1148. p&iacute;&ccedil;&acirc;&Eacute;∞&euml; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;
r r
r r
c
⋅
&Ccedil;
&ecirc;
=
∇
&times;
c
⋅ &Ccedil;p I
∫
∫∫
(
`
)
p
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CHAPTER 9. INTEGRAL CALCULUS
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
r
c(&ntilde; I &oacute; I &ograve; ) = m(&ntilde; I &oacute; I &ograve; )I n(&ntilde; I &oacute; I &ograve; )I o(&ntilde; I &oacute; I &ograve; )
&aacute;&euml; ~ &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; &iuml;&Uuml;&ccedil;&euml;&Eacute; &Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&euml; mI nI ~&aring;&Ccedil; o &Uuml;~&icirc;&Eacute;
&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &eacute;~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;I
r
r
r
&aacute;
&agrave;
&acirc;
r ∂
∂
∂  ∂o ∂n  r  ∂m ∂o  r  ∂n ∂m  r
&aacute; + 
∇&times;c=
=
−
−
− &acirc;
&agrave; +
∂&ntilde; ∂&ntilde; ∂&ntilde;  ∂&oacute; ∂&ograve;   ∂&ograve; ∂&ntilde;   ∂&ntilde; ∂&oacute; 
m n o
r
r
&aacute;&euml; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&auml; &ccedil;&Ntilde; c I ~&auml;&euml;&ccedil; &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil; &Aring;&igrave;&ecirc;&auml; c K
q&Uuml;&Eacute; &euml;&oacute;&atilde;&Auml;&ccedil;&auml;
∫
&aacute;&aring;&Ccedil;&aacute;&Aring;~&iacute;&Eacute;&euml; &iacute;&Uuml;~&iacute; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; &aacute;&euml; &iacute;~&acirc;&Eacute;&aring; &ccedil;&icirc;&Eacute;&ecirc;
~ &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil; &Aring;&igrave;&ecirc;&icirc;&Eacute;K
1149. p&iacute;&ccedil;&acirc;&Eacute;∞&euml; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde; &aacute;&aring; `&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; c&ccedil;&ecirc;&atilde;
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve;
`
 ∂o ∂n 
 ∂n ∂m 
∂m ∂o 
&Ccedil;&oacute;&Ccedil;&ograve; +  −
= ∫∫ 
−
− &Ccedil;&ntilde;&Ccedil;&oacute;
&Ccedil;&ograve;&Ccedil;&ntilde; + 
∂&oacute; ∂&ograve; 
 ∂&ograve; ∂&ntilde; 
 ∂&ntilde; ∂&oacute; 
p 
1150. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ^&ecirc;&Eacute;~
^ = ∫∫ &Ccedil;p
p
1151. f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &aacute;&euml; &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;
r
r
r
r
&ecirc; (&igrave;I &icirc; ) = &ntilde; (&igrave;I &icirc; )&aacute; + &oacute; (&igrave;I &icirc; ) &agrave; + &ograve; (&igrave;I &icirc; )&acirc; I
&iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~ &aacute;&euml;
r
r
∂&ecirc; ∂&ecirc;
&times; &Ccedil;&igrave;&Ccedil;&icirc; I
^ = ∫∫
∂&igrave; ∂&icirc;
a( &igrave; I &icirc; )
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; a(&igrave;I &icirc; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;&ccedil;&atilde;~&aacute;&aring; &iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &ecirc; (&igrave;I &icirc; ) &aacute;&euml;
&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;K
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CHAPTER 9. INTEGRAL CALCULUS
1152. f&Ntilde; p &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Eacute;&ntilde;&eacute;&auml;&aacute;&Aring;&aacute;&iacute;&auml;&oacute; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &ograve; (&ntilde; I &oacute; ) I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; ~&ecirc;&Eacute;~ &aacute;&euml;
O
 ∂&ograve;   ∂&ograve; 
^ = ∫∫ N +   +   &Ccedil;&ntilde;&Ccedil;&oacute; I
 ∂&ntilde;   ∂&oacute; 
a( &ntilde; I &oacute; )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; a(&ntilde; I &oacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &ccedil;&aring;&iacute;&ccedil; &iacute;&Uuml;&Eacute; &ntilde;&oacute;&eacute;&auml;~&aring;&Eacute;K
O
1153. j~&euml;&euml; &ccedil;&Ntilde; ~ p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;
&atilde; = ∫∫ &micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &micro;(&ntilde; I &oacute; I &ograve; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &atilde;~&euml;&euml; &eacute;&Eacute;&ecirc; &igrave;&aring;&aacute;&iacute; ~&ecirc;&Eacute;~ E&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;FK
1154. `&Eacute;&aring;&iacute;&Eacute;&ecirc; &ccedil;&Ntilde; j~&euml;&euml; &ccedil;&Ntilde; ~ p&Uuml;&Eacute;&auml;&auml;
j &oacute;&ograve;
j &ntilde;&oacute;
j
I &oacute; = &ntilde;&ograve; I &ograve; =
I
&ntilde;=
&atilde;
&atilde;
&atilde;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
j &oacute;&ograve; = ∫∫ &ntilde;&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I
p
j &ntilde;&ograve; = ∫∫ &oacute;&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I
p
j &ntilde;&oacute; = ∫∫ &ograve;&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p
p
~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&ecirc;&euml;&iacute; &atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute; &eacute;&auml;~&aring;&Eacute;&euml; &ntilde; = M I
&oacute; = M I &ograve; = M I &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;K &micro;(&ntilde; I &oacute; I &ograve; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K
1155. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; f&aring;&Eacute;&ecirc;&iacute;&aacute;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute; E&ccedil;&ecirc; &ograve; = M FI &oacute;&ograve;-&eacute;&auml;~&aring;&Eacute;
E &ntilde; = M FI ~&aring;&Ccedil; &ntilde;&ograve;-&eacute;&auml;~&aring;&Eacute; E &oacute; = M F
f &ntilde;&oacute; = ∫∫ &ograve; O&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I
p
f &oacute;&ograve; = ∫∫ &ntilde; O&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I
p
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CHAPTER 9. INTEGRAL CALCULUS
f &ntilde;&ograve; = ∫∫ &oacute; O&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p K
p
1156. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml; &ccedil;&Ntilde; f&aring;&Eacute;&ecirc;&iacute;&aacute;~ ~&Auml;&ccedil;&igrave;&iacute; &iacute;&Uuml;&Eacute; &ntilde;-~&ntilde;&aacute;&euml;I &oacute;-~&ntilde;&aacute;&euml;I ~&aring;&Ccedil; &ograve;-~&ntilde;&aacute;&euml;
f &ntilde; = ∫∫ (&oacute; O + &ograve; O )&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I
p
f &oacute; = ∫∫ (&ntilde; O + &ograve; O )&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I
p
f&ograve; = ∫∫ (&ntilde; O + &oacute; O )&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p K
p
1157. s&ccedil;&auml;&igrave;&atilde;&Eacute; &ccedil;&Ntilde; ~ p&ccedil;&auml;&aacute;&Ccedil; _&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil; &Auml;&oacute; ~ `&auml;&ccedil;&euml;&Eacute;&Ccedil; p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;
s=
N
&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; + &oacute;&Ccedil;&ntilde;&Ccedil;&ograve; + &ograve;&Ccedil;&ntilde;&Ccedil;&oacute;
P ∫∫
p
1158. d&ecirc;~&icirc;&aacute;&iacute;~&iacute;&aacute;&ccedil;&aring;~&auml; c&ccedil;&ecirc;&Aring;&Eacute;
r
r
&ecirc;
c = d&atilde; ∫∫ &micro;(&ntilde; I &oacute; I &ograve; ) P &Ccedil;p I
&ecirc;
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &atilde; &aacute;&euml; ~ &atilde;~&euml;&euml; ~&iacute; ~ &eacute;&ccedil;&aacute;&aring;&iacute; &ntilde; M I &oacute; M I &ograve; M &ccedil;&igrave;&iacute;&euml;&aacute;&Ccedil;&Eacute; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;I
r
&ecirc; = &ntilde; − &ntilde; M I &oacute; − &oacute; M I&ograve; − &ograve;M I
&micro;(&ntilde; I &oacute; I &ograve; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;I
~&aring;&Ccedil; d &aacute;&euml; &Ouml;&ecirc;~&icirc;&aacute;&iacute;~&iacute;&aacute;&ccedil;&aring;~&auml; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K
1159. m&ecirc;&Eacute;&euml;&euml;&igrave;&ecirc;&Eacute; c&ccedil;&ecirc;&Aring;&Eacute;
r
r r
c = ∫∫ &eacute;(&ecirc; )&Ccedil;p I
p
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&Eacute;&euml;&euml;&igrave;&ecirc;&Eacute; &eacute;(&ecirc; ) ~&Aring;&iacute;&euml; &ccedil;&aring; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; p &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
r
&iacute;&Uuml;&Eacute; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &ecirc; K
1160. c&auml;&igrave;&aacute;&Ccedil; c&auml;&igrave;&ntilde; E~&Aring;&ecirc;&ccedil;&euml;&euml; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; pF
r r r
Φ = ∫∫ &icirc; (&ecirc; ) ⋅ &Ccedil;p I
p
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r r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &icirc; (&ecirc; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &Ntilde;&auml;&igrave;&aacute;&Ccedil; &icirc;&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;K
1161. j~&euml;&euml; c&auml;&igrave;&ntilde; E~&Aring;&ecirc;&ccedil;&euml;&euml; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; pF
r r r
Φ = ∫∫ ρ&icirc; (&ecirc; ) ⋅ &Ccedil;p I
p
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; c = ρ&icirc; &aacute;&euml; &iacute;&Uuml;&Eacute; &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc; &Ntilde;&aacute;&Eacute;&auml;&Ccedil;I ρ &aacute;&euml; &iacute;&Uuml;&Eacute; &Ntilde;&auml;&igrave;&aacute;&Ccedil; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;K
1162. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; `&Uuml;~&ecirc;&Ouml;&Eacute;
n = ∫∫ σ(&ntilde; I &oacute; )&Ccedil;p I
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; σ(&ntilde; I &oacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &Aring;&Uuml;~&ecirc;&Ouml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;K
1163. d~&igrave;&euml;&euml;∞ i~&iuml;
q&Uuml;&Eacute; &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&auml;&igrave;&ntilde; &iacute;&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml; ~&aring;&oacute; &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute; &aacute;&euml; &eacute;&ecirc;&ccedil;&eacute;&ccedil;&ecirc;&iacute;&aacute;&ccedil;&aring;~&auml;
&iacute;&ccedil; &iacute;&Uuml;&Eacute; &Aring;&Uuml;~&ecirc;&Ouml;&Eacute; n &Eacute;&aring;&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;
r r n
Φ = ∫∫ b ⋅ &Ccedil;p = I
εM
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
Φ &aacute;&euml; &iacute;&Uuml;&Eacute; &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&auml;&igrave;&ntilde;I
r
b &aacute;&euml; &iacute;&Uuml;&Eacute; &atilde;~&Ouml;&aring;&aacute;&iacute;&igrave;&Ccedil;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; &euml;&iacute;&ecirc;&Eacute;&aring;&Ouml;&iacute;&Uuml;I
c
ε M = UIUR &times; NM −NO
&aacute;&euml; &eacute;&Eacute;&ecirc;&atilde;&aacute;&iacute;&iacute;&aacute;&icirc;&aacute;&iacute;&oacute; &ccedil;&Ntilde; &Ntilde;&ecirc;&Eacute;&Eacute; &euml;&eacute;~&Aring;&Eacute;K
&atilde;
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c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &ccedil;&aring;&Eacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W &oacute;I &eacute;I &egrave;I &igrave;I &Ouml;I &Uuml;I dI eI &ecirc;I &ograve;
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;&euml; E&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;FW &ntilde;I &oacute;
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&iuml;&ccedil; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &Ntilde; (&ntilde; I &oacute; ) I j(&ntilde; I &oacute; ) I k(&ntilde; I &oacute; )
&Ccedil;&oacute;
c&aacute;&ecirc;&euml;&iacute; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;W &oacute; ′ I &igrave;′ I &oacute;&amp; I
I&pound;
&Ccedil;&iacute;
&Ccedil; Of
p&Eacute;&Aring;&ccedil;&aring;&Ccedil; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;W &oacute;′′ I &amp;&oacute;&amp; I O I &pound;
&Ccedil;&iacute;
O
∂&igrave; ∂ &igrave;
m~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;W
I
I&pound;
∂&iacute; ∂&ntilde; O
k~&iacute;&igrave;&ecirc;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
m~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;W &oacute; N I &oacute; &eacute;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &acirc;I &iacute;I `I `N I ` O I &eacute;I &egrave;I α I β
o&ccedil;&ccedil;&iacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;W λN I λ O
q&aacute;&atilde;&Eacute;W &iacute;
q&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute;W qI p
m&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W m(&iacute; )
j~&euml;&euml; &ccedil;&Ntilde; ~&aring; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;W &atilde;
p&iacute;&aacute;&Ntilde;&Ntilde;&aring;&Eacute;&euml;&euml; &ccedil;&Ntilde; ~ &euml;&eacute;&ecirc;&aacute;&aring;&Ouml;W &acirc;
a&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &atilde;~&euml;&euml; &Ntilde;&ecirc;&ccedil;&atilde; &Eacute;&egrave;&igrave;&aacute;&auml;&aacute;&Auml;&ecirc;&aacute;&igrave;&atilde;W &oacute;
^&atilde;&eacute;&auml;&aacute;&iacute;&igrave;&Ccedil;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;W ^
c&ecirc;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&oacute;W ω
a~&atilde;&eacute;&aacute;&aring;&Ouml; &Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;W γ
m&Uuml;~&euml;&Eacute; ~&aring;&Ouml;&auml;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;W δ
^&aring;&Ouml;&igrave;&auml;~&ecirc; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;W θ
m&Eacute;&aring;&Ccedil;&igrave;&auml;&igrave;&atilde; &auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W i
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^&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &Ouml;&ecirc;~&icirc;&aacute;&iacute;&oacute;W &Ouml;
`&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute;W f
o&Eacute;&euml;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;W o
f&aring;&Ccedil;&igrave;&Aring;&iacute;~&aring;&Aring;&Eacute;W i
`~&eacute;~&Aring;&aacute;&iacute;~&aring;&Aring;&Eacute;W `
10.1 First Order Ordinary Differential
Equations
1164. i&aacute;&aring;&Eacute;~&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;
&Ccedil;&oacute;
+ &eacute;(&ntilde; )&oacute; = &egrave;(&ntilde; ) K
&Ccedil;&ntilde;
q&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
∫ &igrave;(&ntilde; )&egrave;(&ntilde; )&Ccedil;&ntilde; + ` I
&oacute;=
&igrave; (&ntilde; )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&igrave;(&ntilde; ) = &Eacute;&ntilde;&eacute; ∫ &eacute;(&ntilde; )&Ccedil;&ntilde; K
(
)
1165. p&Eacute;&eacute;~&ecirc;~&Auml;&auml;&Eacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;
&Ccedil;&oacute;
= &Ntilde; (&ntilde; I &oacute; ) = &Ouml; (&ntilde; )&Uuml;(&oacute; )
&Ccedil;&ntilde;
q&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
&Ccedil;&oacute;
∫ &Uuml;(&oacute; ) = ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde; + ` I
&ccedil;&ecirc;
e(&oacute; ) = d(&ntilde; ) + ` K
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1166. e&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;
&Ccedil;&oacute;
= &Ntilde; (&ntilde; I &oacute; ) &aacute;&euml; &Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;I &aacute;&Ntilde;
&Ccedil;&ntilde;
&iacute;&Uuml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &Ntilde; (&ntilde; I &oacute; ) &aacute;&euml; &Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;I &iacute;&Uuml;~&iacute; &aacute;&euml;
&Ntilde; (&iacute;&ntilde; I &iacute;&oacute; ) = &Ntilde; (&ntilde; I &oacute; ) K
q&Uuml;&Eacute; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
q&Uuml;&Eacute; &euml;&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring; &ograve; =
&oacute;
E&iacute;&Uuml;&Eacute;&aring; &oacute; = &ograve;&ntilde; F &auml;&Eacute;~&Ccedil;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &euml;&Eacute;&eacute;~&ecirc;~&Auml;&auml;&Eacute;
&ntilde;
&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
&Ccedil;&ograve;
&ntilde;
+ &ograve; = &Ntilde; (NI &ograve; ) K
&Ccedil;&ntilde;
1167. _&Eacute;&ecirc;&aring;&ccedil;&igrave;&auml;&auml;&aacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
&Ccedil;&oacute;
+ &eacute;(&ntilde; )&oacute; = &egrave;(&ntilde; )&oacute; &aring; K
&Ccedil;&ntilde;
q&Uuml;&Eacute; &euml;&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring; &ograve; = &oacute; N−&aring; &auml;&Eacute;~&Ccedil;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &auml;&aacute;&aring;&Eacute;~&ecirc; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
&Ccedil;&ograve;
+ (N − &aring; )&eacute;(&ntilde; ) &ograve; = (N − &aring; )&egrave;(&ntilde; ) K
&Ccedil;&ntilde;
1168. o&aacute;&Aring;&Aring;~&iacute;&aacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
&Ccedil;&oacute;
= &eacute;(&ntilde; ) + &egrave;(&ntilde; ) &oacute; + &ecirc;(&ntilde; ) &oacute; O
&Ccedil;&ntilde;
f&Ntilde; ~ &eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &oacute; N &aacute;&euml; &acirc;&aring;&ccedil;&iuml;&aring;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &Aring;~&aring; &Auml;&Eacute; &ccedil;&Auml;&iacute;~&aacute;&aring;&Eacute;&Ccedil; &iuml;&aacute;&iacute;&Uuml; &iacute;&Uuml;&Eacute; &Uuml;&Eacute;&auml;&eacute; &ccedil;&Ntilde; &euml;&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;
N
&ograve;=
I &iuml;&Uuml;&aacute;&Aring;&Uuml; &auml;&Eacute;~&Ccedil;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&ecirc;&euml;&iacute; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc; &auml;&aacute;&aring;&Eacute;~&ecirc; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
&oacute; − &oacute;N
&Ccedil;&ograve;
= −[&egrave;(&ntilde; ) + O&oacute; N&ecirc;(&ntilde; )] &ograve; − &ecirc;(&ntilde; ) K
&Ccedil;&ntilde;
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1169. b&ntilde;~&Aring;&iacute; ~&aring;&Ccedil; k&ccedil;&aring;&Eacute;&ntilde;~&Aring;&iacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;
q&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
j(&ntilde; I &oacute; )&Ccedil;&ntilde; + k(&ntilde; I &oacute; )&Ccedil;&oacute; = M
&aacute;&euml; &Aring;~&auml;&auml;&Eacute;&Ccedil; &Eacute;&ntilde;~&Aring;&iacute; &aacute;&Ntilde;
∂j ∂k
=
I
∂&oacute; ∂&ntilde;
~&aring;&Ccedil; &aring;&ccedil;&aring;&Eacute;&ntilde;~&Aring;&iacute; &ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute;K
q&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
∫ j(&ntilde; I &oacute; )&Ccedil;&ntilde; + ∫ k(&ntilde; I &oacute; )&Ccedil;&oacute; = ` K
1170. o~&Ccedil;&aacute;&ccedil;~&Aring;&iacute;&aacute;&icirc;&Eacute; a&Eacute;&Aring;~&oacute;
&Ccedil;&oacute;
= − &acirc;&oacute; I
&Ccedil;&iacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &oacute; (&iacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; ~&atilde;&ccedil;&igrave;&aring;&iacute; &ccedil;&Ntilde; &ecirc;~&Ccedil;&aacute;&ccedil;~&Aring;&iacute;&aacute;&icirc;&Eacute; &Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute; ~&iacute; &iacute;&aacute;&atilde;&Eacute; &iacute;I &acirc;
&aacute;&euml; &iacute;&Uuml;&Eacute; &ecirc;~&iacute;&Eacute; &ccedil;&Ntilde; &Ccedil;&Eacute;&Aring;~&oacute;K
q&Uuml;&Eacute; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
&oacute; (&iacute; ) = &oacute; M &Eacute; − &acirc;&iacute; I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; &oacute; M = &oacute; (M) &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&aacute;&iacute;&aacute;~&auml; ~&atilde;&ccedil;&igrave;&aring;&iacute;K
1171. k&Eacute;&iuml;&iacute;&ccedil;&aring;∞&euml; i~&iuml; &ccedil;&Ntilde; `&ccedil;&ccedil;&auml;&aacute;&aring;&Ouml;
&Ccedil;q
= −&acirc; (q − p ) I
&Ccedil;&iacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; q(&iacute; ) &aacute;&euml; &iacute;&Uuml;&Eacute; &iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute; &ccedil;&Ntilde; ~&aring; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute; ~&iacute; &iacute;&aacute;&atilde;&Eacute; &iacute;I p &aacute;&euml; &iacute;&Uuml;&Eacute;
&iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&igrave;&ecirc;&ecirc;&ccedil;&igrave;&aring;&Ccedil;&aacute;&aring;&Ouml; &Eacute;&aring;&icirc;&aacute;&ecirc;&ccedil;&aring;&atilde;&Eacute;&aring;&iacute;I &acirc; &aacute;&euml; ~ &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K
q&Uuml;&Eacute; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
q(&iacute; ) = p + (qM − p) &Eacute; −&acirc;&iacute; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; qM = q(M) &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&aacute;&iacute;&aacute;~&auml; &iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute; ~&iacute;
&iacute;&aacute;&atilde;&Eacute; &iacute; = M K
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1172. m&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring; a&oacute;&aring;~&atilde;&aacute;&Aring;&euml; Ei&ccedil;&Ouml;&aacute;&euml;&iacute;&aacute;&Aring; j&ccedil;&Ccedil;&Eacute;&auml;F
&Ccedil;m
m

= &acirc;m N −  I
&Ccedil;&iacute;
 j
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; m(&iacute; ) &aacute;&euml; &eacute;&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring; ~&iacute; &iacute;&aacute;&atilde;&Eacute; &iacute;I &acirc; &aacute;&euml; ~ &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;I
j &aacute;&euml; ~ &auml;&aacute;&atilde;&aacute;&iacute;&aacute;&aring;&Ouml; &euml;&aacute;&ograve;&Eacute; &Ntilde;&ccedil;&ecirc; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring;K
q&Uuml;&Eacute; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
jmM
m(&iacute; ) =
I &iuml;&Uuml;&Eacute;&ecirc;&Eacute; mM = m(M) &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&aacute;&iacute;&aacute;~&auml; &eacute;&ccedil;&eacute;&igrave;mM + (j − mM )&Eacute; − &acirc;&iacute;
&auml;~&iacute;&aacute;&ccedil;&aring; ~&iacute; &iacute;&aacute;&atilde;&Eacute; &iacute; = M K
10.2 Second Order Ordinary Differential
Equations
1173. e&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml; i&aacute;&aring;&Eacute;~&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &iuml;&aacute;&iacute;&Uuml; `&ccedil;&aring;&euml;&iacute;~&aring;&iacute; `&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;
&oacute;′′ + &eacute;&oacute;′ + &egrave;&oacute; = M K
q&Uuml;&Eacute; &Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
λO + &eacute;λ + &egrave; = M K
f&Ntilde; λN ~&aring;&Ccedil; λ O ~&ecirc;&Eacute; &Ccedil;&aacute;&euml;&iacute;&aacute;&aring;&Aring;&iacute; &ecirc;&Eacute;~&auml; &ecirc;&ccedil;&ccedil;&iacute;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring;
&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
&oacute; = `N&Eacute; λN&ntilde; + ` O&Eacute; λ O &ntilde; I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
`N ~&aring;&Ccedil; ` O ~&ecirc;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;K
&eacute;
f&Ntilde; λN = λ O = − I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
O
&oacute; = (`N + ` O &ntilde; )&Eacute;
&eacute;
− &ntilde;
O
K
f&Ntilde; λN ~&aring;&Ccedil; λ O ~&ecirc;&Eacute; &Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W
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λN = α + β &aacute; I λ O = α − β &aacute; I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
Q&egrave; − &eacute; O
&eacute;
α=− I β=
I
O
O
&iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
&oacute; = &Eacute; α&ntilde; (`N &Aring;&ccedil;&euml; β &ntilde; + ` O &euml;&aacute;&aring; β &ntilde; ) K
1174. f&aring;&Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml; i&aacute;&aring;&Eacute;~&ecirc; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &iuml;&aacute;&iacute;&Uuml; `&ccedil;&aring;&euml;&iacute;~&aring;&iacute;
`&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;
&oacute;′′ + &eacute;&oacute;′ + &egrave;&oacute; = &Ntilde; (&ntilde; ) K
q&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
&oacute; = &oacute; &eacute; + &oacute; &Uuml; I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&oacute; &eacute; &aacute;&euml; ~ &eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &aacute;&aring;&Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
~&aring;&Ccedil; &oacute; &Uuml; &aacute;&euml; &iacute;&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; ~&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&Eacute;&Ccedil; &Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; E&euml;&Eacute;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&Eacute;&icirc;&aacute;&ccedil;&igrave;&euml; &iacute;&ccedil;&eacute;&aacute;&Aring; NNTPFK
f&Ntilde; &iacute;&Uuml;&Eacute; &ecirc;&aacute;&Ouml;&Uuml;&iacute; &euml;&aacute;&Ccedil;&Eacute; &Uuml;~&euml; &iacute;&Uuml;&Eacute; &Ntilde;&ccedil;&ecirc;&atilde;
&Ntilde; (&ntilde; ) = &Eacute; α&ntilde; (mN (&ntilde; )&Aring;&ccedil;&euml; β &ntilde; + mN (&ntilde; )&euml;&aacute;&aring; β&ntilde; ) I
&iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &oacute; &eacute; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
&oacute; &eacute; = &ntilde; &acirc; &Eacute; α&ntilde; (oN (&ntilde; )&Aring;&ccedil;&euml; β&ntilde; + o O (&ntilde; )&euml;&aacute;&aring; β &ntilde; ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&auml;&oacute;&aring;&ccedil;&atilde;&aacute;~&auml;&euml; oN (&ntilde; ) ~&aring;&Ccedil; o O (&ntilde; ) &Uuml;~&icirc;&Eacute; &iacute;&ccedil; &Auml;&Eacute; &Ntilde;&ccedil;&igrave;&aring;&Ccedil;
&Auml;&oacute; &igrave;&euml;&aacute;&aring;&Ouml; &iacute;&Uuml;&Eacute; &atilde;&Eacute;&iacute;&Uuml;&ccedil;&Ccedil; &ccedil;&Ntilde; &igrave;&aring;&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&Eacute;&Ccedil; &Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;K
• f&Ntilde; α + β &aacute; &aacute;&euml; &aring;&ccedil;&iacute; ~ &ecirc;&ccedil;&ccedil;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;I &iacute;&Uuml;&Eacute;&aring;
&iacute;&Uuml;&Eacute; &eacute;&ccedil;&iuml;&Eacute;&ecirc; &acirc; = M I
• f&Ntilde; α + β &aacute; &aacute;&euml; ~ &euml;&aacute;&atilde;&eacute;&auml;&Eacute; &ecirc;&ccedil;&ccedil;&iacute;I &iacute;&Uuml;&Eacute;&aring; &acirc; = N I
• f&Ntilde; α + β &aacute; &aacute;&euml; ~ &Ccedil;&ccedil;&igrave;&Auml;&auml;&Eacute; &ecirc;&ccedil;&ccedil;&iacute;I &iacute;&Uuml;&Eacute;&aring; &acirc; = O K
1175. a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &iuml;&aacute;&iacute;&Uuml; &oacute; j&aacute;&euml;&euml;&aacute;&aring;&Ouml;
&oacute;′′ = &Ntilde; (&ntilde; I &oacute;′) K
p&Eacute;&iacute; &igrave; = &oacute; ′ K q&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &aring;&Eacute;&iuml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring; &euml;~&iacute;&aacute;&euml;&Ntilde;&aacute;&Eacute;&Ccedil; &Auml;&oacute; &icirc; &aacute;&euml;
&igrave;′ = &Ntilde; (&ntilde; I &igrave; ) I
&iuml;&Uuml;&aacute;&Aring;&Uuml; &aacute;&euml; ~ &Ntilde;&aacute;&ecirc;&euml;&iacute; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;K
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1176. a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &iuml;&aacute;&iacute;&Uuml; &ntilde; j&aacute;&euml;&euml;&aacute;&aring;&Ouml;
&oacute;′′ = &Ntilde; (&oacute; I &oacute;′) K
p&Eacute;&iacute; &igrave; = &oacute; ′ K p&aacute;&aring;&Aring;&Eacute;
&Ccedil;&igrave; &Ccedil;&igrave; &Ccedil;&oacute;
&Ccedil;&igrave;
&oacute;′′ =
=
=&igrave; I
&Ccedil;&ntilde; &Ccedil;&oacute; &Ccedil;&ntilde;
&Ccedil;&oacute;
&iuml;&Eacute; &Uuml;~&icirc;&Eacute;
&Ccedil;&igrave;
&igrave;
= &Ntilde; (&oacute; I &igrave; ) I
&Ccedil;&oacute;
&iuml;&Uuml;&aacute;&Aring;&Uuml; &aacute;&euml; ~ &Ntilde;&aacute;&ecirc;&euml;&iacute; &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;K
1177. c&ecirc;&Eacute;&Eacute; r&aring;&Ccedil;~&atilde;&eacute;&Eacute;&Ccedil; s&aacute;&Auml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;
q&Uuml;&Eacute; &atilde;&ccedil;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; ~ j~&euml;&euml; &ccedil;&aring; ~ p&eacute;&ecirc;&aacute;&aring;&Ouml; &aacute;&euml; &Ccedil;&Eacute;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil; &Auml;&oacute; &iacute;&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
&atilde;&amp;&oacute;&amp; + &acirc;&oacute; = M I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&atilde; &aacute;&euml; &iacute;&Uuml;&Eacute; &atilde;~&euml;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;I
&acirc; &aacute;&euml; &iacute;&Uuml;&Eacute; &euml;&iacute;&aacute;&Ntilde;&Ntilde;&aring;&Eacute;&euml;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&eacute;&ecirc;&aacute;&aring;&Ouml;I
&oacute; &aacute;&euml; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &atilde;~&euml;&euml; &Ntilde;&ecirc;&ccedil;&atilde; &Eacute;&egrave;&igrave;&aacute;&auml;&aacute;&Auml;&ecirc;&aacute;&igrave;&atilde;K
q&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
&oacute; = ^ &Aring;&ccedil;&euml;(ωM &iacute; − δ ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
^ &aacute;&euml; &iacute;&Uuml;&Eacute; ~&atilde;&eacute;&auml;&aacute;&iacute;&igrave;&Ccedil;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;I
ωM &aacute;&euml; &iacute;&Uuml;&Eacute; &Ntilde;&igrave;&aring;&Ccedil;~&atilde;&Eacute;&aring;&iacute;~&auml; &Ntilde;&ecirc;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&oacute;I &iacute;&Uuml;&Eacute; &eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil; &aacute;&euml; q =
δ &aacute;&euml; &eacute;&Uuml;~&euml;&Eacute; ~&aring;&Ouml;&auml;&Eacute; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;K
q&Uuml;&aacute;&euml; &aacute;&euml; ~&aring; &Eacute;&ntilde;~&atilde;&eacute;&auml;&Eacute; &ccedil;&Ntilde; &euml;&aacute;&atilde;&eacute;&auml;&Eacute; &Uuml;~&ecirc;&atilde;&ccedil;&aring;&aacute;&Aring; &atilde;&ccedil;&iacute;&aacute;&ccedil;&aring;K
1178. c&ecirc;&Eacute;&Eacute; a~&atilde;&eacute;&Eacute;&Ccedil; s&aacute;&Auml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;
&atilde;&amp;&oacute;&amp; + γ&oacute;&amp; + &acirc;&oacute; = M I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
γ &aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;~&atilde;&eacute;&aacute;&aring;&Ouml; &Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;K
q&Uuml;&Eacute;&ecirc;&Eacute; ~&ecirc;&Eacute; P &Aring;~&euml;&Eacute;&euml; &Ntilde;&ccedil;&ecirc; &iacute;&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;W
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Oπ
I
ωM
CHAPTER 10. DIFFERENTIAL EQUATIONS
`~&euml;&Eacute; NK γ O &gt; Q&acirc;&atilde; E&ccedil;&icirc;&Eacute;&ecirc;&Ccedil;~&atilde;&eacute;&Eacute;&Ccedil;F
&oacute; (&iacute; ) = ^&Eacute; λN&iacute; + _&Eacute; λ O&iacute; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
λN =
− γ − γ O − Q &acirc;&atilde;
− γ + γ O − Q &acirc;&atilde;
I λO =
K
O&atilde;
O&atilde;
`~&euml;&Eacute; OK γ O = Q&acirc;&atilde; E&Aring;&ecirc;&aacute;&iacute;&aacute;&Aring;~&auml;&auml;&oacute; &Ccedil;~&atilde;&eacute;&Eacute;&Ccedil;F
&oacute; (&iacute; ) = (^ + _&iacute; )&Eacute; λ&iacute; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
γ
λ=−
K
O&atilde;
`~&euml;&Eacute; PK γ O &lt; Q&acirc;&atilde; E&igrave;&aring;&Ccedil;&Eacute;&ecirc;&Ccedil;~&atilde;&eacute;&Eacute;&Ccedil;F
&oacute; (&iacute; ) = &Eacute;
−
γ
&iacute;
O&atilde;
^ &Aring;&ccedil;&euml;(ω&iacute; − δ ) I &iuml;&Uuml;&Eacute;&ecirc;&Eacute;
ω = Q&acirc;&atilde; − γ O K
1179. p&aacute;&atilde;&eacute;&auml;&Eacute; m&Eacute;&aring;&Ccedil;&igrave;&auml;&igrave;&atilde;
&Ccedil; Oθ &Ouml;
+ θ=MI
&Ccedil;&iacute; O i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; θ &aacute;&euml; &iacute;&Uuml;&Eacute; ~&aring;&Ouml;&igrave;&auml;~&ecirc; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;I i &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&Eacute;&aring;&Ccedil;&igrave;&auml;&igrave;&atilde;
&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;I &Ouml; &aacute;&euml; &iacute;&Uuml;&Eacute; ~&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &Ouml;&ecirc;~&icirc;&aacute;&iacute;&oacute;K
q&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &Ntilde;&ccedil;&ecirc; &euml;&atilde;~&auml;&auml; ~&aring;&Ouml;&auml;&Eacute;&euml; θ &aacute;&euml;
i
&Ouml;
θ(&iacute; ) = θ&atilde;~&ntilde; &euml;&aacute;&aring;
&iacute; I &iacute;&Uuml;&Eacute; &eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil; &aacute;&euml; q = Oπ
K
&Ouml;
i
1180. oi` `&aacute;&ecirc;&Aring;&igrave;&aacute;&iacute;
&Ccedil; Of
&Ccedil;f N
i O + o + f = s′(&iacute; ) = ωb M &Aring;&ccedil;&euml;(ω&iacute; ) I
&Ccedil;&iacute;
&Ccedil;&iacute; `
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&iuml;&Uuml;&Eacute;&ecirc;&Eacute; f &aacute;&euml; &iacute;&Uuml;&Eacute; &Aring;&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute; &aacute;&aring; ~&aring; oi` &Aring;&aacute;&ecirc;&Aring;&igrave;&aacute;&iacute; &iuml;&aacute;&iacute;&Uuml; ~&aring; ~&Aring; &icirc;&ccedil;&auml;&iacute;~&Ouml;&Eacute;
&euml;&ccedil;&igrave;&ecirc;&Aring;&Eacute; s(&iacute; ) = b M &euml;&aacute;&aring;(ω&iacute; ) K
q&Uuml;&Eacute; &Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml; &euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring; &aacute;&euml;
f(&iacute; ) = `N&Eacute; &ecirc;N&iacute; + ` O&Eacute; &ecirc;O&iacute; + ^ &euml;&aacute;&aring;(ω&iacute; − ϕ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
Qi
− o &plusmn; oO −
` I
&ecirc; NI O =
Oi
ωb M
^=
I
O
N
 O

O O
 iω −  + o ω
`


N 
 iω
ϕ = ~&ecirc;&Aring;&iacute;~&aring;
−
I
 o o`ω 
`N I ` O ~&ecirc;&Eacute; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml; &Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&aacute;&aring;&Ouml; &ccedil;&aring; &aacute;&aring;&aacute;&iacute;&aacute;~&auml; &Aring;&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;&euml;K
10.3. Some Partial Differential Equations
1181. q&Uuml;&Eacute; i~&eacute;&auml;~&Aring;&Eacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
∂ O&igrave; ∂ O&igrave;
+
=M
∂&ntilde; O ∂&oacute; O
~&eacute;&eacute;&auml;&aacute;&Eacute;&euml; &iacute;&ccedil; &eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml; &Eacute;&aring;&Eacute;&ecirc;&Ouml;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &igrave;(&ntilde; I &oacute; ) &Ntilde;&ccedil;&ecirc; ~ &Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute; &Ntilde;&ccedil;&ecirc;&Aring;&Eacute; &Ntilde;&aacute;&Eacute;&auml;&Ccedil; &aacute;&aring; &iacute;&Uuml;&Eacute; &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K m~&ecirc;&iacute;&aacute;~&auml; &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&aacute;&euml; &iacute;&oacute;&eacute;&Eacute; ~&ecirc;&Eacute; &Aring;~&auml;&auml;&Eacute;&Ccedil; &Eacute;&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;K
1182. q&Uuml;&Eacute; e&Eacute;~&iacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
∂ O &igrave; ∂ O &igrave; ∂&igrave;
+
=
∂&ntilde; O ∂&oacute; O ∂&iacute;
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~&eacute;&eacute;&auml;&aacute;&Eacute;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute; &Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring; &igrave;(&ntilde; I &oacute; ) &aacute;&aring; &iacute;&Uuml;&Eacute; &ntilde;&oacute;&eacute;&auml;~&aring;&Eacute; &iuml;&Uuml;&Eacute;&aring; &Uuml;&Eacute;~&iacute; &aacute;&euml; ~&auml;&auml;&ccedil;&iuml;&Eacute;&Ccedil; &iacute;&ccedil; &Ntilde;&auml;&ccedil;&iuml; &Ntilde;&ecirc;&ccedil;&atilde; &iuml;~&ecirc;&atilde; ~&ecirc;&Eacute;~&euml; &iacute;&ccedil; &Aring;&ccedil;&ccedil;&auml;
&ccedil;&aring;&Eacute;&euml;K q&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&aacute;&euml; &iacute;&oacute;&eacute;&Eacute; ~&ecirc;&Eacute; &Aring;~&auml;&auml;&Eacute;&Ccedil; &eacute;~&ecirc;~&Auml;&ccedil;&auml;&aacute;&Aring;K
1183. q&Uuml;&Eacute; t~&icirc;&Eacute; b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;
∂ O&igrave; ∂ O&igrave; ∂ O&igrave;
+
=
∂&ntilde; O ∂&oacute; O ∂&iacute; O
~&eacute;&eacute;&auml;&aacute;&Eacute;&euml; &iacute;&ccedil; &iacute;&Uuml;&Eacute; &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute; &igrave;(&ntilde; I &oacute; ) &ccedil;&Ntilde; &icirc;&aacute;&Auml;&ecirc;~&iacute;&aacute;&aring;&Ouml; &atilde;&Eacute;&atilde;&Auml;&ecirc;~&aring;&Eacute;&euml;
~&aring;&Ccedil; &ccedil;&iacute;&Uuml;&Eacute;&ecirc; &iuml;~&icirc;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;K q&Uuml;&Eacute; &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml; &ccedil;&Ntilde; &iacute;&Uuml;&aacute;&euml; &iacute;&oacute;&eacute;&Eacute; ~&ecirc;&Eacute;
&Aring;~&auml;&auml;&Eacute;&Ccedil; &Uuml;&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;K
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11.1 Arithmetic Series
f&aring;&aacute;&iacute;&aacute;~&auml; &iacute;&Eacute;&ecirc;&atilde;W ~N
k&iacute;&Uuml; &iacute;&Eacute;&ecirc;&atilde;W ~ &aring;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute; &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring; &euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;&aacute;&icirc;&Eacute; &iacute;&Eacute;&ecirc;&atilde;&euml;W &Ccedil;
k&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&Eacute;&ecirc;&atilde;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W &aring;
p&igrave;&atilde; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&ecirc;&euml;&iacute; &aring; &iacute;&Eacute;&ecirc;&atilde;&euml;W p&aring;
1184. ~ &aring; = ~ &aring; −N + &Ccedil; = ~ &aring; −O + O&Ccedil; = K = ~N + (&aring; − N)&Ccedil;
1185. ~N + ~ &aring; = ~ O + ~ &aring;−N = K = ~ &aacute; + ~ &aring; +N−&aacute;
1186. ~ &aacute; =
~ &aacute; −N + ~ &aacute; +N
O
1187. p&aring; =
~N + ~ &aring;
O~ + (&aring; − N)&Ccedil;
⋅&aring; = N
⋅&aring;
O
O
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11.2 Geometric Series
f&aring;&aacute;&iacute;&aacute;~&auml; &iacute;&Eacute;&ecirc;&atilde;W ~N
k&iacute;&Uuml; &iacute;&Eacute;&ecirc;&atilde;W ~ &aring;
`&ccedil;&atilde;&atilde;&ccedil;&aring; &ecirc;~&iacute;&aacute;&ccedil;W &egrave;
k&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&Eacute;&ecirc;&atilde;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W &aring;
p&igrave;&atilde; &ccedil;&Ntilde; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&ecirc;&euml;&iacute; &aring; &iacute;&Eacute;&ecirc;&atilde;&euml;W p&aring;
p&igrave;&atilde; &iacute;&ccedil; &aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&oacute;W p
1188. ~ &aring; = &egrave;~ &aring; −N = ~N&egrave; &aring; −N
1189. ~N ⋅ ~ &aring; = ~ O ⋅ ~ &aring; −N = K = ~ &aacute; ⋅ ~ &aring; +N−&aacute;
1190. ~ &aacute; = ~ &aacute; −N ⋅ ~ &aacute; +N
~ &aring; &egrave; − ~N ~N (&egrave; &aring; − N)
=
1191. p&aring; =
&egrave; −N
&egrave; −N
~N
&aring; →∞
N− &egrave;
c&ccedil;&ecirc; &egrave; &lt; N I &iacute;&Uuml;&Eacute; &euml;&igrave;&atilde; p &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml; ~&euml; &aring; → ∞ K
1192. p = &auml;&aacute;&atilde; p&aring; =
11.3 Some Finite Series
k&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&Eacute;&ecirc;&atilde;&euml; &aacute;&aring; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W &aring;
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1193. N + O + P + K + &aring; =
&aring;(&aring; + N)
O
1194. O + Q + S + K + O&aring; = &aring;(&aring; + N)
1195. N + P + R + K + (O&aring; − N) = &aring; O
1196. &acirc; + (&acirc; + N) + (&acirc; + O) + K + (&acirc; + &aring; − N) =
1197. NO + OO + PO + K + &aring; O =
&aring;(O&acirc; + &aring; − N)
O
&aring;(&aring; + N)(O&aring; + N)
S
 &aring;(&aring; + N) 
1198. NP + OP + PP + K + &aring;P = 
 O 
O
&aring;(Q&aring; O − N)
1199. N + P + R + K + (O&aring; − N) =
P
O
O
O
O
1200. NP + PP + RP + K + (O&aring; − N) = &aring; O (O&aring; O − N)
P
N N N
N
1201. N + + + + K + &aring; + K = O
O Q U
O
1202.
N
N
N
N
+
+
+K+
+K = N
N⋅ O O ⋅ P P ⋅ Q
&aring;(&aring; + N)
1203. N +
N N N
N
+ + +K+
+K = &Eacute;
(&aring; − N)&gt;
N&gt; O&gt; P&gt;
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11.4 Infinite Series
p&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute;W {~ &aring; }
c&aacute;&ecirc;&euml;&iacute; &iacute;&Eacute;&ecirc;&atilde;W ~N
k&iacute;&Uuml; &iacute;&Eacute;&ecirc;&atilde;W ~ &aring;
1204. f&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; p&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
∑~
&aring; =N
&aring;
= ~N + ~ O + K + ~ &aring; + K
1205. k&iacute;&Uuml; m~&ecirc;&iacute;&aacute;~&auml; p&igrave;&atilde;
&aring;
p&aring; = ∑ ~ &aring; = ~N + ~ O + K + ~ &aring;
&aring; =N
1206. `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; f&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; p&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
∑~
&aring; =N
&aring;
= i I &aacute;&Ntilde; &auml;&aacute;&atilde; p&aring; = i
&aring; →∞
1207. k&iacute;&Uuml; q&Eacute;&ecirc;&atilde; q&Eacute;&euml;&iacute;
∞
•
f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
∑~
&aring; =N
•
&aring;
&aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I &iacute;&Uuml;&Eacute;&aring; &auml;&aacute;&atilde; ~ &aring; = M K
&aring;→∞
f&Ntilde; &auml;&aacute;&atilde; ~ &aring; ≠ M I &iacute;&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml; &aacute;&euml; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
&aring;→∞
11.5 Properties of Convergent Series
∞
`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; p&Eacute;&ecirc;&aacute;&Eacute;&euml;W
∑ ~&aring; = ^ I
&aring; =N
∞
∑&Auml;
&aring; =N
&aring;
=_
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &Aring;
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CHAPTER 11. SERIES
∞
∞
∞
&aring; =N
&aring; =N
&aring; =N
1208. ∑ (~ &aring; + &Auml; &aring; ) = ∑ ~ &aring; + ∑ &Auml; &aring; = ^ + _
1209.
∞
∑ &Aring;~
&aring; =N
∞
&aring;
= &Aring;∑ ~ &aring; = &Aring;^ K
&aring; =N
11.6 Convergence Tests
1210. q&Uuml;&Eacute; `&ccedil;&atilde;&eacute;~&ecirc;&aacute;&euml;&ccedil;&aring; q&Eacute;&euml;&iacute;
∞
i&Eacute;&iacute;
∑ ~ &aring; ~&aring;&Ccedil;
&aring; =N
∞
•
f&Ntilde;
∑&Auml;
&aring; =N
∞
•
f&Ntilde;
∞
∑&Auml;
&aring; =N
&aring;
&Auml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml; &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute; M &lt; ~ &aring; ≤ &Auml;&aring; &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &aring;K
∞
&aring;
&aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &iacute;&Uuml;&Eacute;&aring;
∑~
&aring; =N
∑ ~ &aring; &aacute;&euml; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &iacute;&Uuml;&Eacute;&aring;
&aring; =N
&aring;
&aacute;&euml; ~&auml;&euml;&ccedil; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
∞
∑&Auml;
&aring; =N
&aring;
&aacute;&euml; ~&auml;&euml;&ccedil; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
1211. q&Uuml;&Eacute; i&aacute;&atilde;&aacute;&iacute; `&ccedil;&atilde;&eacute;~&ecirc;&aacute;&euml;&ccedil;&aring; q&Eacute;&euml;&iacute;
∞
i&Eacute;&iacute;
∑ ~ &aring; ~&aring;&Ccedil;
&aring; =N
∞
∑&Auml;
&aring; =N
&aring;
&Auml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml; &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute; ~ &aring; ~&aring;&Ccedil; &Auml;&aring; ~&ecirc;&Eacute; &eacute;&ccedil;&euml;&aacute;-
&iacute;&aacute;&icirc;&Eacute; &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &aring;K
•
•
∞
∞
~&aring;
&lt; ∞ &iacute;&Uuml;&Eacute;&aring; ∑ ~ &aring; ~&aring;&Ccedil; ∑ &Auml;&aring; ~&ecirc;&Eacute; &Eacute;&aacute;&iacute;&Uuml;&Eacute;&ecirc; &Auml;&ccedil;&iacute;&Uuml;
&aring; →∞ &Auml;
&aring; =N
&aring; =N
&aring;
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &ccedil;&ecirc; &Auml;&ccedil;&iacute;&Uuml; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
∞
∞
~
f&Ntilde; &auml;&aacute;&atilde; &aring; = M &iacute;&Uuml;&Eacute;&aring; ∑ &Auml;&aring; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &aacute;&atilde;&eacute;&auml;&aacute;&Eacute;&euml; &iacute;&Uuml;~&iacute; ∑ ~ &aring; &aacute;&euml;
&aring; →∞ &Auml;
&aring; =N
&aring; =N
&aring;
~&auml;&euml;&ccedil; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
f&Ntilde; M &lt; &auml;&aacute;&atilde;
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~&aring;
= ∞ &iacute;&Uuml;&Eacute;&aring;
&aring; →∞ &Auml;
&aring;
~&auml;&euml;&ccedil; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
f&Ntilde; &auml;&aacute;&atilde;
•
∞
∑ &Auml;&aring; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &aacute;&atilde;&eacute;&auml;&aacute;&Eacute;&euml; &iacute;&Uuml;~&iacute;
&aring; =N
∞
∑~
&aring; =N
&aring;
&aacute;&euml;
1212. &eacute;-&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
&eacute;-&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
N
∑&aring;
&aring; =N
&eacute;
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml; &Ntilde;&ccedil;&ecirc; &eacute; &gt; N ~&aring;&Ccedil; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml; &Ntilde;&ccedil;&ecirc;
M &lt; &eacute; ≤NK
1213. q&Uuml;&Eacute; f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; q&Eacute;&euml;&iacute;
i&Eacute;&iacute; &Ntilde; (&ntilde; ) &Auml;&Eacute; ~ &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring; &iuml;&Uuml;&aacute;&Aring;&Uuml; &aacute;&euml; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;I &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;I ~&aring;&Ccedil;
&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml; &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &ntilde; ≥ N K q&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
∑ &Ntilde; (&aring;) = &Ntilde; (N) + &Ntilde; (O) + &Ntilde; (P) + K + &Ntilde; (&aring;) + K
&aring; =N
∞
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml; &aacute;&Ntilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml;I ~&aring;&Ccedil; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml; &aacute;&Ntilde;
N
&aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; → ∞ ~&euml; &aring; → ∞ K
N
1214. q&Uuml;&Eacute; o~&iacute;&aacute;&ccedil; q&Eacute;&euml;&iacute;
∞
i&Eacute;&iacute;
∑~
&aring; =N
•
•
•
&aring;
&Auml;&Eacute; ~ &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml; &iuml;&aacute;&iacute;&Uuml; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &iacute;&Eacute;&ecirc;&atilde;&euml;K
~ &aring; +N
&lt; N &iacute;&Uuml;&Eacute;&aring;
&aring; →∞ ~
&aring;
f&Ntilde; &auml;&aacute;&atilde;
∞
∑~
&aring; =N
&aring;
&aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
∞
~ &aring; +N
&gt; N &iacute;&Uuml;&Eacute;&aring; ∑ ~ &aring; &aacute;&euml; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
&aring; →∞ ~
&aring; =N
&aring;
∞
~
f&Ntilde; &auml;&aacute;&atilde; &aring; +N = N &iacute;&Uuml;&Eacute;&aring; ∑ ~ &aring; &atilde;~&oacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute; &ccedil;&ecirc; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute; ~&aring;&Ccedil;
&aring; →∞ ~
&aring; =N
&aring;
&iacute;&Uuml;&Eacute; &ecirc;~&iacute;&aacute;&ccedil; &iacute;&Eacute;&euml;&iacute; &aacute;&euml; &aacute;&aring;&Aring;&ccedil;&aring;&Aring;&auml;&igrave;&euml;&aacute;&icirc;&Eacute;X &euml;&ccedil;&atilde;&Eacute; &ccedil;&iacute;&Uuml;&Eacute;&ecirc; &iacute;&Eacute;&euml;&iacute;&euml; &atilde;&igrave;&euml;&iacute; &Auml;&Eacute;
&igrave;&euml;&Eacute;&Ccedil;K
f&Ntilde; &auml;&aacute;&atilde;
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1215. q&Uuml;&Eacute; o&ccedil;&ccedil;&iacute; q&Eacute;&euml;&iacute;
∞
∑~
i&Eacute;&iacute;
&aring; =N
•
•
•
&aring;
&Auml;&Eacute; ~ &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml; &iuml;&aacute;&iacute;&Uuml; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &iacute;&Eacute;&ecirc;&atilde;&euml;K
∞
f&Ntilde; &auml;&aacute;&atilde; &aring; ~ &aring; &lt; N &iacute;&Uuml;&Eacute;&aring;
∑~
&aring;
&aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
f&Ntilde; &auml;&aacute;&atilde; &aring; ~ &aring; &gt; N &iacute;&Uuml;&Eacute;&aring;
∑~
&aring;
&aacute;&euml; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
f&Ntilde; &auml;&aacute;&atilde; &aring; ~ &aring; = N &iacute;&Uuml;&Eacute;&aring;
∑~
&aring; →∞
&aring; →∞
&aring; →∞
&aring; =N
∞
&aring; =N
∞
&aring; =N
&aring;
&atilde;~&oacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute; &ccedil;&ecirc; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;I &Auml;&igrave;&iacute;
&aring;&ccedil; &Aring;&ccedil;&aring;&Aring;&auml;&igrave;&euml;&aacute;&ccedil;&aring; &Aring;~&aring; &Auml;&Eacute; &Ccedil;&ecirc;~&iuml;&aring; &Ntilde;&ecirc;&ccedil;&atilde; &iacute;&Uuml;&aacute;&euml; &iacute;&Eacute;&euml;&iacute;K
11.7 Alternating Series
1216. q&Uuml;&Eacute; ^&auml;&iacute;&Eacute;&ecirc;&aring;~&iacute;&aacute;&aring;&Ouml; p&Eacute;&ecirc;&aacute;&Eacute;&euml; q&Eacute;&euml;&iacute; Ei&Eacute;&aacute;&Auml;&aring;&aacute;&ograve;∞&euml; q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;F
i&Eacute;&iacute; {~ &aring; } &Auml;&Eacute; ~ &euml;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml; &euml;&igrave;&Aring;&Uuml; &iacute;&Uuml;~&iacute;
~ &aring;+N &lt; ~ &aring; &Ntilde;&ccedil;&ecirc; ~&auml;&auml; &aring;K
&auml;&aacute;&atilde; ~ &aring; = M K
&aring;→∞
∞
q&Uuml;&Eacute;&aring; &iacute;&Uuml;&Eacute; ~&auml;&iacute;&Eacute;&ecirc;&aring;~&iacute;&aacute;&aring;&Ouml; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
∑ (− N) ~
&aring; =N
∞
&aring;
&aring;
~&aring;&Ccedil;
∑ (− N)
&aring; =N
&aring; −N
~&aring;
&Auml;&ccedil;&iacute;&Uuml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;K
1217. ^&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;
∞
•
^ &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
∑~
&aring; =N
&aring;
&aacute;&euml; ~&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;&auml;&oacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &aacute;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
∑~
&aring; =N
&aring;
&aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
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∞
f&Ntilde; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
•
∑~
&aring; =N
&aring;
&aacute;&euml; ~&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;&auml;&oacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &iacute;&Uuml;&Eacute;&aring; &aacute;&iacute; &aacute;&euml; &Aring;&ccedil;&aring;-
&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
1218. `&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;~&auml; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;
∞
^ &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;
∑~
&aring; =N
&aring;
&aacute;&euml; &Aring;&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;~&auml;&auml;&oacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &aacute;&Ntilde; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml; &aacute;&euml;
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &Auml;&igrave;&iacute; &aacute;&euml; &aring;&ccedil;&iacute; ~&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;&auml;&oacute; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K
11.8 Power Series
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &ntilde;I &ntilde; M
∞
∑~
m&ccedil;&iuml;&Eacute;&ecirc; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W
&aring;=M
&aring;
&ntilde;&aring; I
∞
∑ ~ (&ntilde; − &ntilde; )
&aring; =M
&aring;
&aring;
M
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;W o
1219. m&ccedil;&iuml;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml; &aacute;&aring; &ntilde;
∞
∑~
&aring;=M
&aring;
&ntilde; &aring; = ~ M + ~N&ntilde; + ~ O &ntilde; O + K + ~ &aring; &ntilde; &aring; + K
1220. m&ccedil;&iuml;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml; &aacute;&aring; (&ntilde; − &ntilde; M )
∞
∑ ~ (&ntilde; − &ntilde; )
&aring;=M
&aring;
&aring;
M
= ~ M + ~ N (&ntilde; − &ntilde; M ) + ~ O ( &ntilde; − &ntilde; M ) + K + ~ &aring; ( &ntilde; − &ntilde; M ) + K
O
1221. f&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; &ccedil;&Ntilde; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;
q&Uuml;&Eacute; &euml;&Eacute;&iacute; &ccedil;&Ntilde; &iacute;&Uuml;&ccedil;&euml;&Eacute; &icirc;~&auml;&igrave;&Eacute;&euml; &ccedil;&Ntilde; &ntilde; &Ntilde;&ccedil;&ecirc; &iuml;&Uuml;&aacute;&Aring;&Uuml; &iacute;&Uuml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
∞
&Ntilde; (&ntilde; ) = ∑ ~ &aring; (&ntilde; − &ntilde; M ) &aacute;&euml; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute; &aacute;&euml; &Aring;~&auml;&auml;&Eacute;&Ccedil; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; &ccedil;&Ntilde;
&aring;
&aring; =M
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;K
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&aring;
CHAPTER 11. SERIES
1222. o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;
f&Ntilde; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; &ccedil;&Ntilde; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute; &aacute;&euml; (&ntilde; M − oI &ntilde; M + o ) &Ntilde;&ccedil;&ecirc; &euml;&ccedil;&atilde;&Eacute;
o ≥ M I &iacute;&Uuml;&Eacute; o &aacute;&euml; &Aring;~&auml;&auml;&Eacute;&Ccedil; &iacute;&Uuml;&Eacute; &ecirc;~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;K f&iacute; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring;
~&euml;
~
N
&ccedil;&ecirc; o = &auml;&aacute;&atilde; &aring; K
o = &auml;&aacute;&atilde;
&aring; →∞ ~
&aring; →∞ &aring; ~
&aring; +N
&aring;
11.9 Differentiation and Integration of Power
Series
`&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W &Ntilde; (&ntilde; )
∞
m&ccedil;&iuml;&Eacute;&ecirc; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W
∑~
&aring;=M
&aring;
&ntilde;&aring;
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
o~&Ccedil;&aacute;&igrave;&euml; &ccedil;&Ntilde; `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;W o
1223. a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; m&ccedil;&iuml;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
i&Eacute;&iacute; &Ntilde; (&ntilde; ) = ∑ ~ &aring; &ntilde; &aring; = ~ M + ~ N&ntilde; + ~ O &ntilde; O + K &Ntilde;&ccedil;&ecirc; &ntilde; &lt; o K
&aring; =M
q&Uuml;&Eacute;&aring;I &Ntilde;&ccedil;&ecirc; &ntilde; &lt; o I &Ntilde; (&ntilde; ) &aacute;&euml; &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;I &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute; &Ntilde; ′(&ntilde; )
&Eacute;&ntilde;&aacute;&euml;&iacute;&euml; ~&aring;&Ccedil;
&Ccedil;
&Ccedil;
&Ccedil;
&Ntilde; ′(&ntilde; ) = ~ M + ~ N&ntilde; + ~ O &ntilde; O + K
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&ntilde;
∞
= ~N + O~ O &ntilde; + P~ P &ntilde; O + K = ∑ &aring;~ &aring; &ntilde; &aring; −N K
&aring; =N
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1224. f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; m&ccedil;&iuml;&Eacute;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
i&Eacute;&iacute; &Ntilde; (&ntilde; ) = ∑ ~ &aring; &ntilde; &aring; = ~ M + ~ N&ntilde; + ~ O &ntilde; O + K &Ntilde;&ccedil;&ecirc; &ntilde; &lt; o K
&aring; =M
q&Uuml;&Eacute;&aring;I &Ntilde;&ccedil;&ecirc; &ntilde; &lt; o I &iacute;&Uuml;&Eacute; &aacute;&aring;&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &Eacute;&ntilde;&aacute;&euml;&iacute;&euml; ~&aring;&Ccedil;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ ~ &Ccedil;&ntilde; + ∫ ~ &ntilde;&Ccedil;&ntilde; + ∫ ~ &ntilde; &Ccedil;&ntilde; + K
O
M
N
= ~ M &ntilde; + ~N
O
∞
&ntilde;O
&ntilde;P
&ntilde; &aring; +N
+ ~O + K = ∑ ~&aring;
+`K
O
P
&aring; +N
&aring; =M
11.10 Taylor and Maclaurin Series
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W &Ntilde; (&ntilde; )
o&Eacute;&atilde;~&aacute;&aring;&Ccedil;&Eacute;&ecirc; &iacute;&Eacute;&ecirc;&atilde;W o &aring;
1225. q~&oacute;&auml;&ccedil;&ecirc; p&Eacute;&ecirc;&aacute;&Eacute;&euml;
∞
&Ntilde; (&ntilde; ) = ∑ &Ntilde; (&aring; ) (~ )
&aring;=M
+
&Ntilde;
(&aring; )
(&ntilde; − ~ )&aring;
&aring;&gt;
= &Ntilde; (~ ) + &Ntilde; ′(~ )(&ntilde; − ~ ) +
(~ )(&ntilde; − ~ )&aring; + o
&aring;&gt;
&aring;
&Ntilde; ′′(~ )(&ntilde; − ~ )
+K
O&gt;
K
1226. q&Uuml;&Eacute; o&Eacute;&atilde;~&aacute;&aring;&Ccedil;&Eacute;&ecirc; ^&Ntilde;&iacute;&Eacute;&ecirc; &aring;HN q&Eacute;&ecirc;&atilde;&euml; &aacute;&euml; &Ouml;&aacute;&icirc;&Eacute;&aring; &Auml;&oacute;
&aring; +N
&Ntilde; (&aring; +N) (ξ )(&ntilde; − ~ )
o&aring; =
I ~&lt;ξ&lt;&ntilde; K
(&aring; + N)&gt;
1227. j~&Aring;&auml;~&igrave;&ecirc;&aacute;&aring; p&Eacute;&ecirc;&aacute;&Eacute;&euml;
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O
CHAPTER 11. SERIES
∞
&Ntilde; (&ntilde; ) = ∑ &Ntilde; (&aring; ) (M )
&aring; =M
&Ntilde; ′′(M )&ntilde; O
&Ntilde; (&aring; ) (M )&ntilde; &aring;
&ntilde;&aring;
= &Ntilde; (M ) + &Ntilde; ′(M )&ntilde; +
+K+
+ o&aring;
&aring;&gt;
O&gt;
&aring;&gt;
11.11 Power Series Expansions for Some
Functions
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &ntilde;
1228. &Eacute; &ntilde; = N + &ntilde; +
1229. ~ &ntilde; = N +
&ntilde;O &ntilde;P
&ntilde;&aring;
+
+K+
+K
O&gt; P&gt;
&aring;&gt;
(&ntilde; &auml;&aring; ~ ) + K
&ntilde; &auml;&aring; ~ (&ntilde; &auml;&aring; ~ ) (&ntilde; &auml;&aring; ~ )
+
+
+K+
N&gt;
O&gt;
P&gt;
&aring;&gt;
O
P
&aring;
(− N) &ntilde; &aring;+N &plusmn; K I − N &lt; &ntilde; ≤ N K
&ntilde; O &ntilde;P &ntilde; Q
1230. &auml;&aring;(N + &ntilde; ) = &ntilde; − + − + K +
O P
Q
&aring; +N
&aring;
1231. &auml;&aring;


N+ &ntilde;
&ntilde;P &ntilde;R &ntilde;T
= O &ntilde; + + + + K I &ntilde; &lt; N K
N− &ntilde;
P
R
T


 &ntilde; − N N  &ntilde; − N P N  &ntilde; − N  R 
+ 
1232. &auml;&aring; &ntilde; = O
 + 
 K I &ntilde; &gt; M K
+
+
+
&ntilde;
N
P
&ntilde;
N
R
&ntilde;
N



 

1233. &Aring;&ccedil;&euml; &ntilde; = N −
(− N) &ntilde; O&aring; &plusmn; K
&ntilde;O &ntilde;Q &ntilde;S
+ − +K+
(O&aring; )&gt;
O&gt; Q&gt; S&gt;
&aring;
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CHAPTER 11. SERIES
(− N) &ntilde; O&aring;+N &plusmn; K
&ntilde;P &ntilde;R &ntilde;T
1234. &euml;&aacute;&aring; &ntilde; = &ntilde; − + − + K +
(O&aring; + N)&gt;
P&gt; R&gt; T&gt;
&aring;
1235. &iacute;~&aring; &ntilde; = &ntilde; +
1236. &Aring;&ccedil;&iacute; &ntilde; =
π
&ntilde; P O&ntilde; R NT &ntilde; T SO&ntilde; V
+
+
+
+KI &ntilde; &lt; K
O
P NR
PNR OUPR

N  &ntilde; &ntilde; P O&ntilde; R O&ntilde; T
− + +
+
+ K I &ntilde; &lt; π K
&ntilde;  P QR VQR QTOR

1237. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = &ntilde; +
&ntilde; &lt;NK
1238. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; =
&ntilde; P N ⋅ P&ntilde; R
N ⋅ P ⋅ RK (O&aring; − N)&ntilde; O&aring; +N
+
+K+
+KI
O⋅P O⋅ Q ⋅R
O ⋅ Q ⋅ SK (O&aring; )(O&aring; + N)

π 
&ntilde; P N ⋅ P&ntilde; R
N ⋅ P ⋅ RK (O&aring; − N)&ntilde; O&aring; +N
−&ntilde; +
+
+K+
+ K I
O 
O⋅P O⋅Q ⋅R
O ⋅ Q ⋅ SK (O&aring; )(O&aring; + N)

&ntilde; &lt;NK
1239. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = &ntilde; −
(− N) &ntilde; O&aring;+N &plusmn; K I &ntilde; ≤ N K
&ntilde;P &ntilde;R &ntilde;T
+ − +K+
P
R T
O&aring; + N
&aring;
1240. &Aring;&ccedil;&euml;&Uuml; &ntilde; = N +
&ntilde;O &ntilde;Q &ntilde;S
&ntilde; O&aring;
+
+
+K+
+K
(O&aring; )&gt;
O&gt; Q&gt; S&gt;
1241. &euml;&aacute;&aring;&Uuml; &ntilde; = &ntilde; +
&ntilde;P &ntilde;R &ntilde;T
&ntilde; O &aring; +N
+
+
+K+
+K
(O&aring; + N)&gt;
P&gt; R&gt; T&gt;
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CHAPTER 11. SERIES
11.12 Binomial Series
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &aring;I &atilde;
o&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &ntilde;
`&ccedil;&atilde;&Auml;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;&euml;W &aring; ` &atilde;
1242. (N + &ntilde; ) = N + &aring;`N&ntilde; + &aring;` O &ntilde; O + K + &atilde; ` &aring; &ntilde; &atilde; + K + &ntilde; &aring;
&aring;
1243. &aring; ` &atilde; =
&aring;(&aring; − N)K[&aring; − (&atilde; − N)]
I &ntilde; &lt;NK
&atilde;&gt;
1244.
N
= N− &ntilde; + &ntilde;O − &ntilde;P + K I &ntilde; &lt; N K
N+ &ntilde;
1245.
N
= N+ &ntilde; + &ntilde;O + &ntilde;P + KI &ntilde; &lt; N K
N− &ntilde;
1246. N + &ntilde; = N +
&ntilde; &ntilde; O N ⋅ P&ntilde; P N ⋅ P ⋅ R&ntilde; Q
−
+
−
+KI &ntilde; ≤ NK
O O⋅Q O⋅Q ⋅S O⋅Q ⋅S⋅U
1247. P N + &ntilde; = N +
&ntilde; N ⋅ O&ntilde; O N ⋅ O ⋅ R&ntilde; P N ⋅ O ⋅ R ⋅ U&ntilde; Q
−
+
−
+KI &ntilde; ≤ NK
P P⋅S
P⋅S⋅V
P ⋅ S ⋅ V ⋅ NO
11.13 Fourier Series
f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&Auml;&auml;&Eacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W &Ntilde; (&ntilde; )
c&ccedil;&igrave;&ecirc;&aacute;&Eacute;&ecirc; &Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;W ~ M I ~ &aring; I &Auml;&aring;
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W &aring;
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CHAPTER 11. SERIES
∞
~
1248. &Ntilde; (&ntilde; ) = M + ∑ (~ &aring; &Aring;&ccedil;&euml; &aring;&ntilde; + &Auml;&aring; &euml;&aacute;&aring; &aring;&ntilde; )
O
&aring; =N
π
N
1249. ~ &aring; = ∫ &Ntilde; (&ntilde; )&Aring;&ccedil;&euml; &aring;&ntilde; &Ccedil;&ntilde;
π −π
π
1250. &Auml;&aring; =
N
&Ntilde; (&ntilde; )&euml;&aacute;&aring; &aring;&ntilde; &Ccedil;&ntilde;
π −∫π
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Chapter 12
Probability
12.1 Permutations and Combinations
m&Eacute;&ecirc;&atilde;&igrave;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml;W &aring; m&atilde;
`&ccedil;&atilde;&Auml;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;&euml;W &aring; ` &atilde;
t&Uuml;&ccedil;&auml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W &aring;I &atilde;
1251. c~&Aring;&iacute;&ccedil;&ecirc;&aacute;~&auml;
&aring;&gt; = N ⋅ O ⋅ PK(&aring; − O)(&aring; − N)&aring;
M&gt; = N
1252. &aring; m&aring; = &aring;&gt;
1253. &aring; m&atilde; =
&aring;&gt;
(&aring; − &atilde; )&gt;
1254. _&aacute;&aring;&ccedil;&atilde;&aacute;~&auml; `&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;
&aring;
&aring;&gt;
&aring;
`&atilde; =   =
 &atilde;  &atilde;&gt; (&aring; − &atilde; )&gt;
1255. &aring; ` &atilde; = &aring; ` &aring; −&atilde;
1256. &aring; ` &atilde; + &aring; ` &atilde; +N = &aring; +N ` &atilde; +N
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1257. &aring; ` M + &aring; `N + &aring; ` O + K + &aring; ` &aring; = O&aring;
1258. m~&euml;&Aring;~&auml;∞&euml; q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;
o&ccedil;&iuml; M
o&ccedil;&iuml; N
o&ccedil;&iuml; O
o&ccedil;&iuml; P
o&ccedil;&iuml; Q
o&ccedil;&iuml; R
o&ccedil;&iuml; S
N
N
N
N
N
N
N
O
P
Q
R
S
N
N
P
S
NM
NR
N
Q
NM
OM
NR
12.2 Probability Formulas
b&icirc;&Eacute;&aring;&iacute;&euml;W ^I _
m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;W m
o~&aring;&Ccedil;&ccedil;&atilde; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W uI vI w
s~&auml;&igrave;&Eacute;&euml; &ccedil;&Ntilde; &ecirc;~&aring;&Ccedil;&ccedil;&atilde; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W &ntilde;I &oacute;I &ograve;
b&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil; &icirc;~&auml;&igrave;&Eacute; &ccedil;&Ntilde; uW &micro;
^&aring;&oacute; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &ecirc;&Eacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W ε
p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil; &Ccedil;&Eacute;&icirc;&aacute;~&iacute;&aacute;&ccedil;&aring;W σ
s~&ecirc;&aacute;~&aring;&Aring;&Eacute;W σ O
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W &Ntilde; (&ntilde; ) I &Ntilde; (&iacute; )
1259. m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; &ccedil;&Ntilde; ~&aring; b&icirc;&Eacute;&aring;&iacute;
&atilde;
m( ^ ) = I
&aring;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&atilde; &aacute;&euml; &iacute;&Uuml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;I
&aring; &aacute;&euml; &iacute;&Uuml;&Eacute; &iacute;&ccedil;&iacute;~&auml; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;K
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N
R
N
S
N
CHAPTER 12. PROBABILITY
1260. o~&aring;&Ouml;&Eacute; &ccedil;&Ntilde; m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; s~&auml;&igrave;&Eacute;&euml;
M ≤ m(^ ) ≤ N
1261. `&Eacute;&ecirc;&iacute;~&aacute;&aring; b&icirc;&Eacute;&aring;&iacute;
m( ^ ) = N
1262. f&atilde;&eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute; b&icirc;&Eacute;&aring;&iacute;
m( ^ ) = M
1263. `&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;
m(^ ) = N − m(^ )
1264. f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; b&icirc;&Eacute;&aring;&iacute;&euml;
m(^ L _ ) = m(^ ) I
m(_ L ^ ) = m(_ )
1265. ^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring; o&igrave;&auml;&Eacute; &Ntilde;&ccedil;&ecirc; f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; b&icirc;&Eacute;&aring;&iacute;&euml;
m(^ ∪ _ ) = m(^ ) + m(_ )
1266. j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring; o&igrave;&auml;&Eacute; &Ntilde;&ccedil;&ecirc; f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute; b&icirc;&Eacute;&aring;&iacute;&euml;
m(^ ∩ _ ) = m(^ ) ⋅ m(_ )
1267. d&Eacute;&aring;&Eacute;&ecirc;~&auml; ^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring; o&igrave;&auml;&Eacute;
m(^ ∪ _ ) = m(^ ) + m(_ ) − m(^ ∩ _ ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
^ ∪ _ &aacute;&euml; &iacute;&Uuml;&Eacute; &igrave;&aring;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &Eacute;&icirc;&Eacute;&aring;&iacute;&euml; ^ ~&aring;&Ccedil; _I
^ ∩ _ &aacute;&euml; &iacute;&Uuml;&Eacute; &aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring; &ccedil;&Ntilde; &Eacute;&icirc;&Eacute;&aring;&iacute;&euml; ^ ~&aring;&Ccedil; _K
1268. `&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;~&auml; m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;
m(^ ∩ _ )
m( ^ L _ ) =
m(_ )
1269. m(^ ∩ _ ) = m(_ ) ⋅ m(^ L _ ) = m(^ ) ⋅ m(_ L ^ )
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CHAPTER 12. PROBABILITY
1270. i~&iuml; &ccedil;&Ntilde; q&ccedil;&iacute;~&auml; m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;
&atilde;
m(^ ) = ∑ m(_ &aacute; )m(^ L _ &aacute; ) I
&aacute; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; _ &aacute; &aacute;&euml; ~ &euml;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; &atilde;&igrave;&iacute;&igrave;~&auml;&auml;&oacute; &Eacute;&ntilde;&Aring;&auml;&igrave;&euml;&aacute;&icirc;&Eacute; &Eacute;&icirc;&Eacute;&aring;&iacute;&euml;K
1271. _~&oacute;&Eacute;&euml;∞ q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;
m(^ L _ ) ⋅ m(_ )
m(_ L ^ ) =
m(^ )
1272. _~&oacute;&Eacute;&euml;∞ c&ccedil;&ecirc;&atilde;&igrave;&auml;~
m(_ ) ⋅ m(^ L _ &aacute; )
m(_ &aacute; L ^ ) = &atilde; &aacute;
I
∑ m(_ &aacute; ) ⋅ m(^ L _ &aacute; )
&acirc; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
_ &aacute; &aacute;&euml; ~ &euml;&Eacute;&iacute; &ccedil;&Ntilde; &atilde;&igrave;&iacute;&igrave;~&auml;&auml;&oacute; &Eacute;&ntilde;&Aring;&auml;&igrave;&euml;&aacute;&icirc;&Eacute; &Eacute;&icirc;&Eacute;&aring;&iacute;&euml; E&Uuml;&oacute;&eacute;&ccedil;&iacute;&Uuml;&Eacute;&euml;&Eacute;&euml;FI
^ &aacute;&euml; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&aring;~&auml; &Eacute;&icirc;&Eacute;&aring;&iacute;I
m(_ &aacute; ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&aacute;&ccedil;&ecirc; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&aacute;&Eacute;&euml;I
m(_ &aacute; L ^ ) ~&ecirc;&Eacute; &iacute;&Uuml;&Eacute; &eacute;&ccedil;&euml;&iacute;&Eacute;&ecirc;&aacute;&ccedil;&ecirc; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&aacute;&Eacute;&euml;K
1273. i~&iuml; &ccedil;&Ntilde; i~&ecirc;&Ouml;&Eacute; k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
p

m &aring; − &micro; ≥ ε  → M ~&euml; &aring; → ∞ I
 &aring;

p

m &aring; − &micro; &lt; ε  → N ~&euml; &aring; → ∞ I
 &aring;

&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
p&aring; &aacute;&euml; &iacute;&Uuml;&Eacute; &euml;&igrave;&atilde; &ccedil;&Ntilde; &ecirc;~&aring;&Ccedil;&ccedil;&atilde; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;I
&aring; &aacute;&euml; &iacute;&Uuml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;K
1274. `&Uuml;&Eacute;&Auml;&oacute;&euml;&Uuml;&Eacute;&icirc; f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;
s(u )
m( u − &micro; ≥ ε ) ≤ O I
ε
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; s(u ) &aacute;&euml; &iacute;&Uuml;&Eacute; &icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute; &ccedil;&Ntilde; uK
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CHAPTER 12. PROBABILITY
1275. k&ccedil;&ecirc;&atilde;~&auml; a&Eacute;&aring;&euml;&aacute;&iacute;&oacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
( &ntilde; −&micro; ) O
−
N
O
ϕ(&ntilde; ) =
&Eacute; Oσ I
σ Oπ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &ntilde; &aacute;&euml; ~ &eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;K
1276. p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil; k&ccedil;&ecirc;&atilde;~&auml; a&Eacute;&aring;&euml;&aacute;&iacute;&oacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
O
N − &ograve;O
ϕ(&ograve; ) =
&Eacute;
Oπ
^&icirc;&Eacute;&ecirc;~&Ouml;&Eacute; &icirc;~&auml;&igrave;&Eacute; &micro; = M I &Ccedil;&Eacute;&icirc;&aacute;~&iacute;&aacute;&ccedil;&aring; σ = N K
Figure 210.
1277. p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil; w s~&auml;&igrave;&Eacute;
u−&micro;
w=
σ
1278. `&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute; k&ccedil;&ecirc;&atilde;~&auml; a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&ntilde;
−
N
c(&ntilde; ) =
&Eacute;
∫
σ Oπ −∞
( &iacute; −&micro; ) O
OσO
&Ccedil;&iacute; I
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CHAPTER 12. PROBABILITY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&ntilde; &aacute;&euml; ~ &eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;I
&iacute; &aacute;&euml; ~ &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;K
 α −&micro; β−&micro;
1279. m(α &lt; u &lt; β ) = c
 − c
I
 σ   σ 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
u &aacute;&euml; &aring;&ccedil;&ecirc;&atilde;~&auml;&auml;&oacute; &Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&Eacute;&Ccedil; &ecirc;~&aring;&Ccedil;&ccedil;&atilde; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;I
c &aacute;&euml; &Aring;&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute; &aring;&ccedil;&ecirc;&atilde;~&auml; &Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;I
m(α &lt; u &lt; β ) &aacute;&euml; &aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;K
ε
1280. m( u − &micro; &lt; ε ) = Oc  I
σ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
u &aacute;&euml; &aring;&ccedil;&ecirc;&atilde;~&auml;&auml;&oacute; &Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&Eacute;&Ccedil; &ecirc;~&aring;&Ccedil;&ccedil;&atilde; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;I
c &aacute;&euml; &Aring;&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute; &aring;&ccedil;&ecirc;&atilde;~&auml; &Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K
1281. `&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute; a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&ntilde;
c(&ntilde; ) = m(u &lt; &ntilde; ) = ∫ &Ntilde; (&iacute; )&Ccedil;&iacute; I
−∞
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute; &aacute;&euml; ~ &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute; &ccedil;&Ntilde; &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;K
1282. _&Eacute;&ecirc;&aring;&ccedil;&igrave;&auml;&auml;&aacute; q&ecirc;&aacute;~&auml;&euml; m&ecirc;&ccedil;&Aring;&Eacute;&euml;&euml;
&micro; = &aring;&eacute; I σ O = &aring;&eacute;&egrave; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&aring; &aacute;&euml; ~ &euml;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute; &ccedil;&Ntilde; &Eacute;&ntilde;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&aring;&iacute;&euml;I
&eacute; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; &ccedil;&Ntilde; &euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml; &ccedil;&Ntilde; &Eacute;~&Aring;&Uuml; &Eacute;&ntilde;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&aring;&iacute;&euml;I
&egrave; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; &ccedil;&Ntilde; &Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;I &egrave; = N − &eacute; K
1283. _&aacute;&aring;&ccedil;&atilde;&aacute;~&auml; a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
 &aring;
&Auml;(&aring;I &eacute;I &egrave; ) =   &eacute; &acirc; &egrave; &aring; − &acirc; I
&acirc;
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CHAPTER 12. PROBABILITY
&micro; = &aring;&eacute; I σ O = &aring;&eacute;&egrave; I
&Ntilde; (&ntilde; ) = (&egrave; + &eacute;&Eacute; &ntilde; ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&aring; &aacute;&euml; &iacute;&Uuml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &iacute;&ecirc;&aacute;~&auml;&euml; &ccedil;&Ntilde; &euml;&Eacute;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;I
&eacute; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; &ccedil;&Ntilde; &euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;I
&egrave; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; &ccedil;&Ntilde; &Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;I &egrave; = N − &eacute; K
&aring;
1284. d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring; a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;
m(q = &agrave;) = &egrave; &agrave;−N&eacute; I
N
&egrave;
&micro; = I σO = O I
&eacute;
&eacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
q &aacute;&euml; &iacute;&Uuml;&Eacute; &Ntilde;&aacute;&ecirc;&euml;&iacute; &euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;&Ntilde;&igrave;&auml; &Eacute;&icirc;&Eacute;&aring;&iacute; &aacute;&euml; &iacute;&Uuml;&Eacute; &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;I
&agrave; &aacute;&euml; &iacute;&Uuml;&Eacute; &Eacute;&icirc;&Eacute;&aring;&iacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;I
&eacute; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; &iacute;&Uuml;~&iacute; ~&aring;&oacute; &ccedil;&aring;&Eacute; &Eacute;&icirc;&Eacute;&aring;&iacute; &aacute;&euml; &euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;&Ntilde;&igrave;&auml;I
&egrave; &aacute;&euml; &iacute;&Uuml;&Eacute; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute; &ccedil;&Ntilde; &Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;I &egrave; = N − &eacute; K
1285. m&ccedil;&aacute;&euml;&euml;&ccedil;&aring; a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;
λ&acirc; −λ
m(u = &acirc; ) ≈ &Eacute; I λ = &aring;&eacute; I
&acirc;&gt;
O
&micro; =λI σ =λ I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
λ &aacute;&euml; &iacute;&Uuml;&Eacute; &ecirc;~&iacute;&Eacute; &ccedil;&Ntilde; &ccedil;&Aring;&Aring;&igrave;&ecirc;&ecirc;&Eacute;&aring;&Aring;&Eacute;I
&acirc; &aacute;&euml; &iacute;&Uuml;&Eacute; &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc; &ccedil;&Ntilde; &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;K
1286. a&Eacute;&aring;&euml;&aacute;&iacute;&oacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Auml;
m(~ ≤ u ≤ &Auml;) = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde;
~
1287. `&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; r&aring;&aacute;&Ntilde;&ccedil;&ecirc;&atilde; a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;
~+&Auml;
N
I &micro;=
I
&Ntilde;=
&Auml;−~
O
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CHAPTER 12. PROBABILITY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Ntilde; &aacute;&euml; &iacute;&Uuml;&Eacute; &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute; &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K
1288. b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml; a&Eacute;&aring;&euml;&aacute;&iacute;&oacute; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
&Ntilde; (&iacute; ) = λ&Eacute; −λ&iacute; I &micro; = λ I σ O = λO
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute; &aacute;&euml; &iacute;&aacute;&atilde;&Eacute;I λ &aacute;&euml; &iacute;&Uuml;&Eacute; &Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute; &ecirc;~&iacute;&Eacute;K
1289. b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml; a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring; c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
c(&iacute; ) = N − &Eacute; −λ&iacute; I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &iacute; &aacute;&euml; &iacute;&aacute;&atilde;&Eacute;I λ &aacute;&euml; &iacute;&Uuml;&Eacute; &Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute; &ecirc;~&iacute;&Eacute;K
1290. b&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil; s~&auml;&igrave;&Eacute; &ccedil;&Ntilde; a&aacute;&euml;&Aring;&ecirc;&Eacute;&iacute;&Eacute; o~&aring;&Ccedil;&ccedil;&atilde; s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;
&aring;
&micro; = b(u ) = ∑ &ntilde; &aacute; &eacute;&aacute; I
&aacute; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &ntilde; &aacute; &aacute;&euml; ~ &eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;I &eacute; &aacute; &aacute;&euml; &aacute;&iacute;&euml; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;K
1291. b&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil; s~&auml;&igrave;&Eacute; &ccedil;&Ntilde; `&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; o~&aring;&Ccedil;&ccedil;&atilde; s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;
∞
&micro; = b(u ) = ∫ &ntilde;&Ntilde; (&ntilde; )&Ccedil;&ntilde;
−∞
1292. m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml; &ccedil;&Ntilde; b&ntilde;&eacute;&Eacute;&Aring;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml;
b(u + v ) = b(u ) + b(v ) I
b(u − v ) = b(u ) − b(v ) I
b(&Aring;u ) = &Aring;b(u ) I
b(uv ) = b(u ) ⋅ b(v ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Aring; &aacute;&euml; ~ &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K
1293. b(u O ) = s(u ) + &micro; O I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&micro; = b(u ) &aacute;&euml; &iacute;&Uuml;&Eacute; &Eacute;&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil; &icirc;~&auml;&igrave;&Eacute;I
s(u ) &aacute;&euml; &iacute;&Uuml;&Eacute; &icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;K
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1294. j~&ecirc;&acirc;&ccedil;&icirc; f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;
b(u )
m(u &gt; &acirc; ) ≤
I
&acirc;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &acirc; &aacute;&euml; &euml;&ccedil;&atilde;&Eacute; &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K
1295. s~&ecirc;&aacute;~&aring;&Aring;&Eacute; &ccedil;&Ntilde; a&aacute;&euml;&Aring;&ecirc;&Eacute;&iacute;&Eacute; o~&aring;&Ccedil;&ccedil;&atilde; s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;
[
]
&aring;
σ O = s(u ) = b (u − &micro; ) = ∑ (&ntilde; &aacute; − &micro; ) &eacute;&aacute; I
O
O
&aacute; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
&ntilde; &aacute; &aacute;&euml; ~ &eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc; &ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;I
&eacute; &aacute; &aacute;&euml; &aacute;&iacute;&euml; &eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;K
1296. s~&ecirc;&aacute;~&aring;&Aring;&Eacute; &ccedil;&Ntilde; `&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml; o~&aring;&Ccedil;&ccedil;&atilde; s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;
[
] ∫ (&ntilde; − &micro;) &Ntilde; (&ntilde; )&Ccedil;&ntilde;
σ = s(u ) = b (u − &micro; ) =
O
O
∞
O
−∞
1297. m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml; &ccedil;&Ntilde; s~&ecirc;&aacute;~&aring;&Aring;&Eacute;
s(u + v ) = s(u ) + s(v ) I
s(u − v ) = s(u ) + s(v ) I
s(u + &Aring; ) = s(u ) I
s(&Aring;u ) = &Aring; O s(u ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute; &Aring; &aacute;&euml; ~ &Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K
1298. p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil; a&Eacute;&icirc;&aacute;~&iacute;&aacute;&ccedil;&aring;
[
a(u ) = s(u ) = b (u − &micro; )
O
]
1299. `&ccedil;&icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;
&Aring;&ccedil;&icirc; (uI v ) = b[(u − &micro;(u ))(v − &micro;(v ))] = b(uv ) − &micro;(u )&micro;(v ) I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
u &aacute;&euml; &ecirc;~&aring;&Ccedil;&ccedil;&atilde; &icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;I
s(u ) &aacute;&euml; &iacute;&Uuml;&Eacute; &icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute; &ccedil;&Ntilde; uI
&micro; &aacute;&euml; &iacute;&Uuml;&Eacute; &Eacute;&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil; &icirc;~&auml;&igrave;&Eacute; &ccedil;&Ntilde; u &ccedil;&ecirc; vK
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1300. `&ccedil;&ecirc;&ecirc;&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;
&Aring;&ccedil;&icirc; (uI v )
I
ρ(uI v ) =
s(u )s(v )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;
s(u ) &aacute;&euml; &iacute;&Uuml;&Eacute; &icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute; &ccedil;&Ntilde; uI
s(v ) &aacute;&euml; &iacute;&Uuml;&Eacute; &icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute; &ccedil;&Ntilde; vK
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```