APÉNDICE IV. LAS ECUACIONES DE MAXWELL EN UN SOLO

Anuncio
567
!
APÉNDICE IV. LAS ECUACIONES DE MAXWELL EN UN SOLO SISTEMA
AXIOMATICO
Por John Auping con base en dos artículos de José Heras Gómez 1713
!
!!!"#$%!&'()*+,%!$+%)%!,+),-!'&.$%#/!
01!!!!0)$.-*2,,+3)/!4&#!%,2&,+-)%#!52)*&6%)$&4%#!*%!7&89%44!:!*%!4&!52%.;&!*%!<-.%)$;!:!*%!!
!!!!!!!!!4&!,-)#%.=&,+3)!*%!4&!,&.>&!
001!!!!?-.624&,+3)!,-=&.+&)$%!*%!4&#!%,2&,+-)%#!*%!7&89%44!:!*%!4&!%,2&,+3)!*%!,-)$+)2+*&*!!
0001!!!@%.+=&,+3)!*%!4&#!%,2&,+-)%#!*%!7&89%44!*%!4&!%,2&,+3)!*%!,-)$+)2+*&*!
0001A1<&#!%,2&,+-)%#!$%)#-.+&4%#!*%!7&89%44!#-)!,-6'&$+B4%#!,-)!4&!*%!,-)$+)2+*&*!
0001C1<&#!*-#!%,2&,+-)%#!$%)#-.+&4%#!#%!*%.+=&)!*%!4&!%,2&,+3)!*%!!,-)$+)2+*&*!
0D1!!!<&#!%,2&,+-)%#!*%!-)*&!*%!4-#!,&6'-#!%4(,$.+,-!:!6&>)($+,-!
D1!!!!E-$%),+&4%#!%4%,$.-6&>)($+,-#!
D01!!!"4!,&.F,$%.!&8+-6F$+,-!*%4!%4%,$.-6&>)%$+#6-!
!
I. INTRODUCCION: LAS ECUACIONES FUNDAMENTALES DE MAXWELL Y
DE LA FUERZA DE LORENTZ Y DE LA CONSERVACIÓN DE LA CARGA
!!!!G+! &,%'$&6-#! 4&! ,-)=%),+3)! $.&*+,+-)&4! *%! %#,.+B+.! 4-#! ,&6'-#! &! *%$%.6+)&.! %)! %4! 4&*-!
+;H2+%.*-! *%! 4&#! %,2&,+-)%#! :! 4&#! 52%)$%#! %#'%,+5+,&*&#! %)! %4! 4&*-! *%.%,I-J! %)$-),%#! 4&! 6&)%.&!
&'.-'+&*&! *%! %#,.+B+.! !"#$ %&"'()$ *%&"%+),*#$ -*$ ."/0*!!$ -*$ !"$ *!*%'()-+,12+%"$ *,$ *!$ *#3"%+)$
'(+-+2*,#+),"!!,-)!%4!G+#$%6&!0)$%.)&,+-)&4!*%!2)+*&*%#!%#!4&!#+>2+%)$%/!
K0"@L!
!!!!!!!!!!!!!!!!!!!!!K<%:!*%!M&2##!%)!%4%,$.+,+*&*L!!
K00"@L!
!!!!!!!!!!!!!!K<%:!*%!+)*2,,+3)!*%!?&.&*&:L!
K000"@L
!!!!!!!!!!!!!!!!!!!!!!!K<%:!*%!M&2##!%)!6&>)%$+#6-L!
K0D"@L!
!!!K<%:!*%!N6'O.%P7&89%44L!!!
!!!*-)*%
Q! ! %#! %4! ,&6'-! %4(,$.+,-Q! ! %4! ,&6'-! 6&>)($+,-Q! ! %#! 4&! *%)#+*&*! *%!
,&.>&!K52),+3)!%#,&4&.LQ!:! !4&!*%)#+*&*!*%!,-..+%)$%!K52),+3)!=%,$-.+&4L1!<&#!%,2&,+-)%#!00!:!000!
#-)! I-6->()%&#! K'-.H2%! $+%)%)! 52%)$%! R! ,%.-LJ! :! 4&! 0! :! 0D! #-)! )-PI-6->()%&#! K'-.H2%! $+%)%)!
52%)$%! )-P,%.-L1! "4! #+#$%6&! 0"@P0D"@! %#! 2)! ,-)S2)$-! *%! %,2&,+-)%#! *+5%.%),+&4%#! '&.,+&4%#! :!
&,-'4&*&#!%)!4-#!,&6'-#!%4(,$.+,-!:!6&>)($+,-1!!
!
")! %4! ,&#-! +6'-.$&)$%! *%! 4&! "#"$%&'(%)%*$+! :! ,+-."%'(%)%*$+J! 4-#! ,&6'-#! :! 52%)$%#! #-)!
+)*%'%)*+%)$%#!*%4!$+%6'-!K
!:!
LJ!:!*%'%)*%)!#-4-!*%!4&#!,--.*%)&*&#!%#'&,+&4%#1!
E-.!4-!$&)$-J!%4!#+#$%6&!0"@P0D"@!#%!$.&)#5-.6&!%)!%4!#+#$%6&!0"GP0D"GJ!*%!4&!#+>2+%)$%!6&)%.&/!
K0"GL!
!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
ATAU
!V-#(!W%.&#J!CXXTJ!YZ&)!7&89%44[#!%H2&$+-)#!B%!-B$&+)%*!5.-6!$I%!,-)$+)2+$:!%H2&$+-)\]!%)/!!"#$%&'())*+,$('-)
+.)/012%&23)=-41!T^!KCXXTL/!_^CP_^TQ!:!V-#(!W%.&#J!!YW-9!$-!-B$&+)!$I%!,-=&.+&)$!5-.6!-5!7&89%44[#!%H2&$+-)#!5.-6!
$I%!,-)$+)2+$:!%H2&$+-)]J!%)/!4,$+5#'()*+,$('-)+.)/012%&2J!=-41!UX!KCXX`L/!ab^Pa^b!
568
!
"##$%&!
!
"###$%&!
!
"#'$%&!
!!!!
!
$(! )*! +,-.! /)! 0(,! .(/,! )*)+12.3,4(516+,! 307! ,*)8,/,! /)! -0! 90)(1):! !"#$ %&'#('! ;,2,! 96()-!
;2<+16+.-:!*,!+,24,!7!*,!+.226)(1)!-.(!+)2.-:!/)!3./.!=0)!*,-!)+0,+6.()-!-.(!1./,-!>.3.45(),-?!
"#%@&!
!
"##%@&!
"###%@&!
"#'%@&!
!
!
!
$
$$A2,/6+6.(,*3)(1):! !"# $%&'("# %!%)$'&*"+,-$.)"# %/# 01"!:! )-! /)+62:! )&,/./$%# 0%# 0&/# "2.&*"/#
.,0%3%,0.%,$%/:!,!-,B)2:!*,-!)+0,+6.()-!/)!+,3;.!"*,-!)+0,+6.()-!/)!C,DE)**&!7!*,!)+0,+6F(!/)!
90)2G,H! ')+1.26,*3)(1):! !"# 41%'5"# 0%# 6&'%,$5! )(12)! )*! +,3;.! )*)+12.3,4(516+.! 7! 0(,! +,24,!
)*5+126+,!-)!2);2)-)(1,!;.2!*,!-6406)(1)!)+0,+6F(IJIK?!
"'&!
")+0,+6F(!/)!*,!90)2G,!/)!L.2)(1G&!
$
!!L,! )+0,+6F(! /)! +.(16(06/,/! "0(! ;26(+6;6.! 3,1)3<16+.&! )-! *,! 2);2)-)(1,+6F(! 3,1)3<16+,! /)! *,!
+.(-)2M,+6F(!/)!*,!+,24,!"0(!;26(+6;6.!9N-6+.&H!L,-!)+0,+6.()-!!"#$$%$!&"#$ $'()*'+,-$*,$%+.,+'/-$
0%$+1-2'-.'0,03!
"'#O&
!
!!!P,/.!=0)!
"'#Q&!
!
:!-)!-640)!=0)?!
!!!")+0,+6F(!/)!*,!+.(-)2M,+6F(!/)!*,!+,24,&!
!
!!!$(! M6-1,! /)! *,! /)3.-12,+6F(! ,(1)26.2! ")+0,+6.()-! /)! C,DE)** ! )+0,+6F(! /)! +.(16(06/,/&! *,!
3,7.2N,!/)!*.-!*6B2.-!6(-6-1)(!)(!/)+62!=0)!*,!+.(-)2M,+6F(!/)!*,!+,24,!")+0,+6F(!/)!+.(16(06/,/&!
(.! )-! 0(! ;26(+6;6.! 6(/);)(/6)(1)! -6(.! 0(,! +.(-)+0)(+6,! /)! *,-! )+0,+6.()-! /)! C,DE)**H! %6(!
)3B,24.:! ;,2,! .B1)()2! *,! )+0,+6F(! /)! O3;)2)RC,DE)**! #'$P:! *.-! *6B2.-! 2)+022)(! ,! *,!
+.(-)2M,+6F(!/)!+,24,H!%)!16)()!)(1.(+)-!0(!,2403)(1.!+62+0*,2?!;,2,!/)26M,2!*,-!)+0,+6.()-!/)!
C,DE)**! 2)+02263.-! ,! *,! +.(-)2M,+6F(! /)! *,! +,24,! 7! ;,2,! /)26M,2! *,! +.(-)2M,+6F(! /)! *,! +,24,!
2)+02263.-!,!*,-!)+0,+6.()-!/)!C,DE)**H!!
$
$
$
$
$
$
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
IJIK
!!P,M6/!S269961>:!!-2410.+2'1-$21$"*%24105-,('+6H!"ITTT&?!UVK!
569
!
II. FORMULACIÓN COVARIANTE DE LAS ECUACIONES DE MAXWELL Y
LA DE CONTINUIDAD
!
!"#!$%&'(!)#)$*+(&%,-.*/$(!0)!1)2/-)!'(+!)#!0/,3/)-*)!*)-0(+!%-*/0/&.*+/$(4!
567
!*)-0(+!1)!$%&'(!)#)$*+(&%,-.*/$(!
!
!!!89(+%!:/)-;!)#!*)-0(+!13%#!1)#!$%&'(!)#)$*+(&%,-.*/$(!0)!$%#$3#%!$(-!!
5<7!
!!!
!!!!!!!1(-1)!!
67!
<7!
E7!
!)0!)#!0=&:(#(!1)!>)?/@A/?/*%;!B3);!'%+*/)-1(!1)!
;!0)!1)2/-)!$(&(4!
;!0/!9%C!3-!-D&)+(!'%+!1)!$(-&3*%$/(-)0!1)!1(0!=-1/$)0!
;!0/!9%C!3-!-D&)+(!/&'%+!1)!1(0!=-1/$)0!
;!)-!*(1(0!#(0!1)&F0!$%0(0!5'(+!)G)&'#(!$3%-1(!)#!&/0&(!=-1/$)!0)!+)'/*)7!
!
5E7
*)-0(+!13%#!1)!$%&'(!)#)$*+(&%,-.*/$(!
!
!!!!H)2/-/&(0!%9(+%!)#!$3%1+/?)$*(+!$%+,%@$(++/)-*)!
5
;$ 7;!!B3)!+)'+)0)-*%!#%!23)-*)!1)#!
6K6L
$%&'(!
I!J-!*)(+)&%!1)#!$F#$3#(!*)-0(+/%# !1/$)!B3)!3-!*)-0(+!%-*/0/&.*+/$(!)-!)#!)0'%$/(@
*/)&'(!1)!M/-N(O0N/!B3)1%!1)*)+&/-%1(!%#!)0')$/2/$%+!#%0!$%-*/1%1)0!4!
5P7!
5L7!
!!C!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
6K6L
! H(-%#1! Q(:)! 56RSP7;! TU)#&9(#*V! *9)(+)&! 2(+! %-*/0C&&)*+/$! 0)$(-1@+%-N! *)-0(+! 2/)#10! %-1! )#)$*+(&%,-)*/0&!
O/*9!&%,-)*/$!&(-('(#)0!"!!)-4!#$%&'()*+,-.&*)/+-0+1234'(4!!?(#I!L<!56RRW74!ELP@ELSX!C!Y(0.!U)+%0;!T8!09(+*!'+((2!
(2!*9)!,)-)+%#/V)1!U)#&9(#*V!*9)(+)&;"!#$%&'()*+,-.&*)/+-0+1234'(4!+?(#I!LS!56RRW74!6LP@6LL!
570
!
"#!$%&'(%#!)#*+&,#,!-./!0(!11#'#!,&2(%3(*)&#!,(!
)8)1&)#9!:&!(0$()&;&)#'<0!1#!$%&'(%#!)#*+&,#,!)<'<!!
-=/
!4!1#!0(35*,#!)#*+&,#,!-6/!(0!5*#!%(1#)&7*!
!!"#$%#&'(")*('+,'-'./("%)0!
!
4!1#!0(35*,#!)#*+&,#,!)<'<>!
!!"#$%#&'(")*,'-'./("%)0*
-?/
!
!!!"#!)<*0(%2#)&7*!,(!1#!)#%3#!0(!%($%(0(*+#!'#+('@+&)#'(*+(!$<%!1#!()5#)&7*!,(!)<*+&*5&,#,>!
!
-AB/!
-C/!
!
!!-
!
/!!!"#$%#&1(*2"()'3&%4*5"*4%*#'()"36%#&1(*5"*4%*#%3.%0!
!
!!!!"#0!()5#)&<*(0!-=/!4!-?/!%($%(0(*+#*!1#!;<%'51#)&7*!)<2#%&#*+(!,(!1#0!()5#)&<*(0!,(!D#EF(11G!
(*!;<%'#!+(*0<%&#1!4!1#!-C/!(0!1#!()5#)&7*!+(*0<%&#1!,(!1#!)<*0(%2#)&7*!,(!1#!)#%3#9!"#!()5#)&7*!
-=/!%($%(0(*+#!1#0!()5#)&<*(0!,(!D#EF(11!&*H<'<3I*(#0!4!1#!()5#)&7*!-?/!1#0!H<'<3I*(#09!!
!
J<*! %(1#)&7*! #1! +(*0<%! #*+&0&'I+%&)<!
(0! $(%+&*(*+(! H#)(%! (1! 0&35&(*+(! )<'(*+#%&<! 3(*(%#19!!
K*!+(*0<%!#%L&+%#%&<!(*!5*!(0$#)&<!,(!)5#+%<!,&'(*0&<*(0!4!+&(*(!M=!)<'$<*(*+(09!:&!(0+(!+(*0<%!
(0! 0&'I+%&)<! (*+<*)(0! 0<1<! MN! ,(! 050! M=! )<'$<*(*+(0! 0<*! &*,($(*,&(*+(09! :&! (1! +(*0<%! (0!
#*+&0&'I+%&)<! (*+<*)(0! 0<1<! =! ,(! 050! M=! )<'$<*(*+(0! ! 0<*! &*,($(*,&(*+(0G! $<%O5(!
!4!1<0!(1('(*+<0!,(!1#!
+%#P#!0<*!*51<0G!#!0#L(%G!
9!Q1!+(*(%!0(&0!)<'$<*(*+(0!&*,($(*,&(*+(0!
(*!5*!(0$#)&<R+&('$<!,(!)5#+%<!,&'(*0&<*(0G!5*!+(*0<%!#*+&0&'I+%&)<!(0!$#%+&)51#%'(*+(!#$+<!$#%#!
%($%(0(*+#%! ! (1! )#'$<! (1I)+%&)<! ! 4! (1! )#'$<! '#3*I+&)<! 9! J#,#! )#'$<! )<*+%&L54(! )<*! S!
)<'$<*(*+(0!&*,($(*,&(*+(09!T(%<G!(1!H()H<!O5(!5*!+(*0<%!#*+&0&'I+%&)<!(*!5*!(0$#)&<R+&('$<!,(!
)5#+%<! ,&'(*0&<*(0! (0! #$+<! $#%#! %($%(0(*+#%! 1<0! )#'$<0! (1I)+%&)<! 4! '#3*I+&)<G! 0<1#'(*+(!
0&3*&;&)#! O5(! (0! !"#$%&'()*! )<*! 1#0! ()5#)&<*(0! ,(! D#EF(11! 4! ;#1+#! +<,#28#! +*,'-%,! (0+(! +(*0<%G!
'.+*$*.+'*.&*#*.&*/,(!1#0!()5#)&<*(0!,(!D#EF(119!
!!!
!!!A(%&;&O5('<0! #H<%#! O5(! 1#! ;<%'51#)&7*! )<2#%&#*+(! ,(! 1#0! ()5#)&<*(0! ,(! D#EF(11! (*! )5#+%<!
,&'(*0&<*(0!-,(1!(0$#)&<R+&('$</!%($%<,5)(!;&(1'(*+(!1#!;<%'51#)&7*!505#1!,(!+%(0!,&'(*0&<*(0!
(0$#)&#1(09! T#%#! ,('<0+%#%! O5(! 1#! ()5#)&7*! +(*0<%&#1! -=/! %($%<,5)(! 1#0! ()5#)&<*(0! ,&;(%(*)&#1(0!
&*H<'<3I*(#0!H#)('<0!
>!
-U/
-MN/
!
!
571
!
"#$%&
!!!!!!!"
&!
!!!'()(!*+,-.-)!/(!-01(023.!"#4&5!616,2,127*6!-.!"8&5!
"99&
!
!
"9:&
!
"9;&
!
!!
!!!<6(.=*!9!-.!"99&5!"9:&!>!"9;&5!*+,-.-7*6?!
"9@&
!
"9A&
!
"98&
!
!
!!!BC)1D(.=*!"9@&5!"9A&!>!"98&5!>!,*7(.=*!-.!01-.,(!E1-!
"#4$%&!
!!!!!!!!!"
5!*+,-.-7*6?!
&!
!
!!B!0*.,2.1(023.!6-!0*7D)*+()F!037*!/(!6-C1.=(!-01(023.!,-.6*)2(/!"G&!0*.,2-.-!/(6!-01(02*.-6!
##!>!###!=-!H(IJ-//K!L-0*)=-7*6!/(!-01(023.!,-.6*)2(/?!
"G&!!
!
!
'()(!*+,-.-)!/(!"###&5!616,2,127*6!-.!"G&!
"9G&
!
"9M&
!
!
!
!
"!###$%&!!!
!!!"
&!
!
'()(!*+,-.-)!"##&5!616,2,127*6!-.!"G&!
5
!N!!!!
572
!
"!
#$%&
!
#'(&!
!!!!!!!!#)*!)*)+),-.! &!
#'$&
!
#''&!
!!!
#'/&!
!
!!#)*!)*)+),-.! &!
#'0&!
!!!#)*!)*)+),-.! &!
!
123!)45246.,)3!)342*27)3!#'(&8!#''&!9!#'0&!3)!:5);),!7):7)3),-27!<)4-.762*+),-)8!+);62,-)!5,!
!:7.;54-.!7.-246.,2*"!
#'=&!
!
!
!!!!!*.!>5)!,.3!;2!*2!3)?5,;2!)45246@,!;)!A2BC)**"!
#DDEF&!!!
!#
&!!
!!
!!!!!!!!!!
!
!
III. DERIVACIÓN DE LAS ECUACIONES DE MAXWELL DE LA ECUACIÓN
DE CONTINUIDAD
!
G.;7H2+.3!I24)7!*2!;)76<246@,!+),46.,2;2!),!)*!)3:246.!-76;6+),36.,2*8!532,;.!<)4-.7)38!:)7.!
)3-)!42+6,.!)3!+59!*27?.J!G.7!)3.8!.:-2+.3!:.7!)*!42+6,.!)3:2462*K-)+:.72*8!),!L.7+2!-),3.762*J!
M2!;)+.3-72+.3!>5)!*23!452-7.!)45246.,)3!;)!A2BC)**!3)!:5);),!)3476N67!4.+.!;.3!)45246.,)3!
-),3.762*)3!),!)*!)3:246.K-6)+:.!;)!A6,O.C3O6!#<P23)!2776N2!3)446@,!DD&J!QI.72!,.3!-.42!;)76<27!!
)3-23!;.3!)45246.,)3!2!:27-67!;)!*2!)45246@,!-),3.762*!;)!4.,-6,56;2;!#3)446@,!DDDJ$&R!;)3:5P38!:272!
,.!42)7!),!!"#$#$%&!'$()$!$$*&;)+.3-72+.3!>5)!)*!-),3.7!;)*!42+:.!)*)4-7.+2?,P-64.!
!#<)7!2!
4.,-6,5246@,! *2! )45246@,! SD&! >5)! 532+.3! :272! **)?27! ;)! *23! ;.3! )45246.,)3! -),3.762*)3! 2! *23!
573
!
"#$%&'!("#$")'*(+!,)-(&(*")$.(+!,(!/$01(..2!+(!,(&)3$2!(*!-'&4$!)*,(5(*,)(*%(!,(!(+%$+!6.%)4$+2!
,(!.$!.(7!,(!8'#.'49:;'&(*%<!=+("")>*!???@AB@!
!
III.1. LAS DOS ECUACIONES TENSORIALES SON COMPATIBLES CON LA
ECUACIÓN TENSORIAL DE CONTINUIDAD
!
!!!C*! !"#$%& '()*"#+%,-& ,%& (./%.+0)& 1(& ,%& .")*(#2%.+0)& 1(& ,%& .%#3%! (+%D! )45.E")%$! (*! .$+!
("#$")'*(+!F'4'GH*($+!,(!/$01(..2!7!(+!"'45$%)9.(!"'*!(..$+I!
!!!!!!!!!!=
=JB
B!
!
!!!!!!K(&'2!.$!!"#$%&'(')'*%*!*'!*'+!,)"(!L#(!.$+!("#$")'*(+!,(!/$01(..!+,-*,.'/%0!,(!.$!!
("#$")>*!,(!"'*%)*#),$,@!C+%$!,(&)3$")>*!+(!..(3$&D!$!"$9'!$!"'*%)*#$")>*@!
!
;'+! '9M(%'+!
! 7!
! &(5&(+(*%$*! ,'+! "#$,&)3("%'&(+2! $! +$9(&! (.! ,(! "#$.L#)(&! 5#*%'! (*! (.!
"$45'! (.("%&'4$G*H%)"'! 7! (.! ,(.! 5#*%'! -#(*%(! ::,'*,(! (+%D! .$! "$&G$::2! &(+5("%)3$4(*%(@!
N("'&,(4'+2!"'*!&(+5("%'!$!(+%'+!'9M(%'+2!OP!&(G.$+!,(.!D.G(9&$!%(*+'&)$.I!
$@:!
9@:!
"@:!
,@:!
(@:!
!(+!#*$!,(&)3$,$!"'*%&$3$&)$*%(Q!
!(+!#*$!,(&)3$,$!"'3$&)$*%(Q!
!(+!#*$!,(&)3$,$!"'3$&)$*%(Q!
!(+!#*$!,(&)3$,$!"'*%&$3$&)$*%(Q!!
!7!!
!
-@:!(*!(.!"$+'!,(!(+%(!(*+$7'2!
!
I!!
!!!!!!=4H%&)"$!R2R2R2:!7!*'!F$7!+#4$!+'9&(!$.5F$B!!
G@:!
F@:!
!(+!#*!%(*+'&!"'*%&$3$&)$*%(!7!
!(+!#*!%(*+'&!"'3$&)$*%(Q!
!(+!.$!,(&)3$,$!"'3$&)$*%(!,(!#*!%(*+'&!"'*%&$3$&)$*%(@!
)@:! ,(-)*)4'+! #*$! -#*")>*!
! =.$! -#*")>*! ,(! S&((*! L#(! ,(5(*,(! ,(! .$! ,)-(&(*")$! ,(!
,'+!5#*%'+B!,(!%$.!4$*(&$!L#(!+(!"#45.$!L#(!%(*G$!,'+!5&'5)(,$,(+2!$!+$9(&!)!7!MQ!(+%$!-#*")>*!(+!
!7!.$+!,'+!5&'5)(,$,(+!+'*I!
M@:!
T@:!
!!!7!
!
574
!
"#$!
!%&!'(&'%!")!*+,-%+)!,&.%/+)"!%0!%"!1("2-%&!!
!!!!!!!3!")!0%/2&')!%"!.,%-*(4!)-5(0!'%! !)!
-#$!!6%(+%-)!'%!7)2008!
!
!0,!
&#$!9%".)!'%!9,+):8!
!!!!!!
!
!3!%0! !0,!
!0,!
;!3!)&<"(/)-%&.%8!!
!3!%0! !0,!
;!
*#$!!=&!"(0!*2&.(0!>)?!@)0.)!>-?4!*('%-(04!(51,)-%&.%4!020.,.2,+! !*(+!
!!!!!!+%:(+')+4!)'%-<04!A2%!
#!
!
B%:(+'%-(0!")!%:2):,C&!.%&0(+,)"!'%!")!:(&0%+1):,C&!'%!")!:)+/)8!
>D?!
!!!!>+%:2%+')8!
#!=0!,-*(+.)&.%!!
!
!
9%+,1)-(0!)-5(0!")'(0!'%!>D?!:(&!
>EF?
8!
!
!
G2".,*",:)-(0!)-5(0!")'(0!'%!>EF?!:(&!
>EH?!
8!
!
!
I@(+)!02-)-(0!)!)-5(0!")'(0!'%!")!%:2):,C&!
>ED?!
8!
!
!
I@(+)!+%%0:+,5,+%-(0!"(0!'(0!.J+-,&(0!'%"!")'(!,KA2,%+'(!'%!>ED?!:(-(!0,/2%8!
>EL?!
!
!!!M+2%5)8!
!
!>NO?!
!
!
!
575
!
!!!!!!!!!!!!"
!!!!!!!!!!!!"
!
!!!#$!%$!&$!
!
!!!'()*+!,)-./0+-)1!2345!6!2375!8+*+!).9:0:*!;+!1/<=/:09:!/>:09/>+>!.?1/,+@A@BC!
2D@5!
!
!
!!!'()*+!-=;9/8;/,+-)1!;)1!9E*-/0)1!,)0!
!:!/09:<*+-)1!1).*:!9)>)!:;!:18+,/)F9/:-8)!!
2D35
C!
!
!
!!!&+>)! G=:! :;! )8:*+>)*!
8=:>:! 1+;/*! >:;! 1/<0)! >:! /09:<*+;H! I:*:-)1! ,+>+! =0)! >:! ;)1! 9*:1!
9E*-/0)1!>:!2D35!8)*!1:8+*+>)H!:-8:J+0>)!,)0!;+!>:;!;+>)!/JG=/:*>)!>:!;+!:,=+,/K0C!
2DD5
!
!!!!!
!!!L+*+!:;!1:<=0>)!9E*-/0)!>:!2D35H!(:-)1!>:!*:,)*>+*!:;!9:)*:-+!>:!M+=11!28=09)!N!+**/.+5$!L)*!
:;!9:)*:-+!>:!M+=11H!;+!/09:<*+;!>:;!:18+,/)F9/:-8)!,=+9*/>/-:01/)0+;!1:!9*+01O)*-+!:0!;+!/09:<*+;!
>:!=0+!1=8:*O/,/:!9*/>/-:01/)0+;$!P)1!9E*-/0)1!>:!:19+!/09:<*+;!G=:!1:!:0,=:09*+0!:09*:!8+*E09:1/1!
!*:8*:1:09+0!,+*<+1!6!,)**/:09:1!O/0/9+1$!';!/09:<*+*!
H!G=:!
:1! =0! 9:01)*! >:! >)1! Q0>/,:1! :0! ,=+9*)! >/-:01/)0:1H! 1).*:!
H! G=:! :1! =0+! 1=8:*O/,/:! >:! 9*:1!
>/-:01/)0:1H! .+R+-)1! :;! 0S-:*)! >:! >/-:01/)0:1! >:! ,=+9*)! +! 9*:1$! %1! ,)-)! 1/! :T9:0>/E*+-)1!
:19+1!,+*<+1!6!,)**/:09:1!O/0/9+1!1).*:!=0+!1=8:*O/,/:!/0O/0/9+H!*+JK0!8)*!;+!,=+;!:19+!/09:<*+;!>:;!
1:<=0>)!9E*-/0)!>:!2D35!*:1=;9+!1:*!,:*)C!
2DU5!
!
!
'()*+!+0+;/J+-)1!;+!9:*,:*+!/09:<*+;!2+8;/,+0>)!;)1!8=09)1!/!6!0!>:!+**/.+5C!
2DV5!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
@A@B
!W)1E!X:*+1!23YY45H!Z[(:!:T+,9!*:;+9/)01(/8!.:9\::0!9(:!>/18;+,:-:09!,=**:09!+0>!9(:!,)0>=,9/)0!,=**:09]H!:0C!!
!!!!"#$%&'()*+,$('-)+.)/012%&2H!I);$!AB!23YY45C!DH!:,=+,/K0!4!
576
!
!
"#$%&'(')#!*++,-!*+.,!/!*+0,-!122341&%&12$#3!*+5,!4#$#!*+6,7!
!
*+6,!
!
!!!!829&'&$#3!!"#$!%&'(#)%$*+&*,-$(*.'#/!#.0)$('#/*,!%&*'%!&7!
*+:,!
!!!!
!
!!!!;<3=&=<&$#3!*+:,!2'!*+6,!>(1(!#%=2'21!*+?,7!
*+?,!
!
!!
!!!!@24#1)2$#37!
*6,!!
!!
!
##!"#$%&%'(") ) *+,) -) *./,0) 1"'23%) 456) 7%) $&8#6&%) 615%189') 36'2"&8%7) (6) :%;<677) *+=./,0) 62)
1"#$%38>76) 1"') 7%) 615%189') (6) 7%) 1"'26&?%189') (6) 7%) 1%&@%) *7%) /,0) 286#$&6) -) 15%'(") 67) 36'2"&)
6')*./,)26)8(6'38A8456)1"#"B)
*A,
!
!
!!!BC!>1#%C2$(!23!D<2!'#!>#)2$#3!()#>=(1!23=(!&)2'=&9&4(4&E'-!(>#/F')#'#3!2'!C(3!24<(4&#'23!)2!
G(HI2CC-! >#1D<2! 4(21J($#3! 2'! <'! $63838") $&8'18$88C) K#)(LJ(! 9(C=(! )21&L(1! 23=2! =2'3#1!
!
&')2>2')&2'=2$2'=2!)2!C(3!24<(4&#'23!)2!G(HI2CCM!
!
!!!N#! $&3$#! >#)2$#3! O(421! 4#'! C(! 32P<')(! 24<(4&E'! =2'3#1&(C-! 23! )24&1-! 4#$>1#%(1! 3<!
4#$>(=&%&C&)()!4#'!
!*2C!=2'3#1!)2!C(!24<(4&E'!A,M!!@24#1)2$#37!!
*+:,!
!!!!
577
!
!!!"#$%&'()*!'(+)*!,'-)*!.)/!
0!
1234
!!!
!
!!!5)/!+'*#!#/!,'!6$)6%#-'-!%7!)+8#/#()*0!
!!!
19:4!
!
!!!;<)$'!+%#/7!*#!*'+#!=>#0!
19?4!!!
!
!
!
!
!!!!@,! 6$%(#$! 8A$(%/)! -#,! ,'-)! -#$#.<)! -#! 19?4! #*! %B>',! ',! %/8#B$'/-)! -#,! ,'-)! -#$#.<)! -#! 19:4C!
D>*8%8>%()*!,'!-%E#$#/.%'!-#,!6$%(#$!F!8#$.#$!8A$(%/)!-#!19?4!#/!19:40!
19G4!
!
!!!!!
!!!5)/!#,!8#)$#('!-#!H'>**7!8$'/*E)$('()*!#,!*#B>/-)!8A$(%/)!-#!19G4C!I)$!#,!8#)$#('!-#!H'>**7!
,'! %/8#B$',! -#,! #*6'.%)J8%#(6)! .>'8$%-%(#/*%)/',! *#! 8$'/*E)$('! #/! ,'! %/8#B$',! -#! >/'! *>6#$E%.%#!
8$%-%(#/*%)/',C!K)*!8A$(%/)*!-#!#*8'!%/8#B$',!=>#!*#!#/.>#/8$'/!#/8$#!6'$A/8#*%*!1 4!$#6$#*#/8'/!
.'$B'*! F! .)$$%#/8#*! E%/%8'*C! ;,! %/8#B$'$! ! *)+$#!
! #*! .)()! *%! #L8#/-%A$'()*! #*8'*! .'$B'*! F!
.)$$%#/8#*!E%/%8'*!*)+$#!>/'!*>6#$E%.%#!%/E%/%8'7!$'MN/!6)$!,'!.>',!#*8'!%/8#B$',!-#,!*#B>/-)!8A$(%/)!
-#!19G4!$#*>,8'!*#$!.#$)0!
1924!
!
!
!!!!I)$!,)!8'/8)7!.)(+%/'/-)!19G4!F!19247!)+8#/#()*0!
1994!
!
!
19O4!
!
!
19P4!
!
!
!
!
578
!
!
!!!"#$%&!'()*+!),!-%(.)%!/0%.(*$!1),!,&1$!1)%)2#$!1)!3445!6!),!7)89*1$!/0%.(*$!1),!,&1$!1)%)2#$!
1)!34:5!7)!&*9,&*+!&7(.(7.$!),!7)89*1$!/0%.(*$!1),!,&1$!1)%)2#$!1)!3445!6!),!-%(.)%!/0%.(*$!1),!
,&1$!1)%)2#$!1)!34;5+!6!),!-%(.)%!/0%.(*$!1),!,&1$!1)%)2#$!1)!34:5!6!),!7)89*1$!/0%.(*$!1),!,&1$!
1)%)2#$!1)!34;5+!1)!.$1$!<9)=!
!
34>5!
!
!
!!?)2$%1).$7!,&!7)89*1&!)29&2(@*!/)*7$%(&,!1)!A&BC),,=!
3>5
!
!
!!!D$.-&%&*1$! 3>5! 6! 34>5! 2$*7/&/&.$7! <9)! ,&! 7)89*1&! )29&2(@*! /)*7$%(&,! 1)! A&BC),,+! &,! (89&,!
<9)!,&!-%(.)%&+!7)!1)%(E&!1)!,&!2$*7)%E&2(@*!1)!,&!2&%8&!7().-%)!6!29&*1$!),!/)*7$%!
)*!34>5!
7)!(1)*/(F(<9)!2$.$=!!
3G5
!
!
!!!H,! -%$',).&! )7+! 2$.$! 6&! 7)! 1(I$! &%%('&+! <9)! *$! -$1).$7! &1$-/&%! )7/&! (1)*/(F(2&2(@*+!
&-$6J*1$*$7! )*! ,&7! )29&2($*)7! 1)! A&BC),,+! -$%<9)! 2&)%K&.$7! )*! 9*!!"#$#$%& !'$()$!$$*&L$1&EK&!
F&,/&!1)%(E&%!)7/)!/)*7$%!
!(*1)-)*1()*/).)*/)!1)!,&7!)29&2($*)7!1)!A&BC),,M!N$%!)I).-,$+!7(!
&1$-/J%&.$7!,&!7(89()*/)!(1)*/(F(2&2(@*!34O5=!
!
34O5
!
!
!!)*/$*2)7+!,&!)29&2(@*!34>5!(.-,(2&%K&=!
34P5!!!!
!
!
+!
579
!
!!"#!$%#"!&'!()*&*!+#,*!*-.*/)0*&(#"!1!&'!*,!%&#!2*!"#,!*$%#$)'&*,!2*!3#-4*""5!6'0*&(#!7*/#,8!
9!"#$%&"%'()#*$#%+),')&'*"*#-&$*$#,"./'0)#'.-!'%"1#+,1"2#,$+13"2#$!$%,1+."4)0,'%"2#*'5$1$),$2#
*$#!"#*$#6"78$!!9:;<;<!=&#!$'&,)2*/#$)>&!#2)$)'&#"!*,!*&('&$*,!&*$*,#/)#?!.#/#!)/!0@,!#""@!2*!"#!
$'0.#()+)")2#2!0#(*0@()$#?!1!A%,()B)$#/!"#!*"*$$)>&!2*!"#,!$'0.'&*&(*,!$'&!+#,*!*&!#/C%0*&(',!
2*!"#!BD,)$#5!
!
III.2. LAS DOS ECUACIONES TENSORIALES SE DERIVAN DE LA ECUACIÓN
TENSORIAL DE LA CONSERVACION DE LA CARGA.
!!!
!!!E'2*0',!./'$*2*/!*&!C*&*/#"!*()F%*(#&2'!"#,!$'0.'&*&(*,!2*"!(*&,'/!
*&!GH<I!$'0'!,)C%*8!
GJKI
!
!!!!
!!!:+#;&$#;&$1$.+2#%+.-1+/"1#$2#!"#2'4&'$),$#'*$),'*"*5!
GJ;I
L
!
!
!!M#0',! #! $'&,)2*/#/! $!# !3.',$# ')*$-$)*'$),$# *$!# ,'$.-+! 2*! "#,! *$%#$)'&*,! GN<I! 1! GH<I! *&!
$'&A%&$)>&! $'&! "#,! "*1*,! 2*! 6'%"'0+! 1! O)'(PQ#R#/(?! .#/#! A%,()B)$#/! *,(#! )2*&()B)$#$)>&5!!
S*$'/2*0',!%&#!R*T!0@,8!
GN<I!
!!!!
!
!!!U*!"#!GN<I?!'+(*&*0',8!
GJVI!
!!!!!1!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
;<;<
! W',X! 7*/#,?! 97'4! ('! '+(#)&! (Y*! $'R#/)#&(! B'/0! 'B! 3#-4*""Z,! *F%#()'&,! B/'0! (Y*! $'&()&%)(1! *F%#()'&:?! *&8!
<&1+-$")#=+&1)"!#+5#>?@2'%2A#R'"5!NK!GVKK[I8!!\J;!
580
!
"#$%!
!!!!
!
!!!!&'()*+',)-!./!*'0./!"1%!/**12/!3)*,4./+/5!
!
"#6%7!"8%!
!
!!!9'!"#$%!:!"#6%!)2;'<',)-5!
"##%!
!!!!9/+)!=4'>!-1!
!!!
>!-'!-104'!=4'!./!'(4/(1?<!"##%!-'!;*/<-3)*,/!'<5!
"#@%!
!!!
!!!'<! +)<+'!
! :!
>! '-! +'(1*>! ./!
+'<-1+/+! +'! (/*0/! '-;A! 'B/.4/+/! '<! '.! C4<;)! 34'<;'! "
;*/<-3)*,/!'<5!
"#F%!
%>! +)<+'! '-;A! ./! (/4-/D! E/! "#@%! -'!
!!!
!!!!
!!!!G<!'.!!!"#$%&'$()&*&()$&(%&')&!'%$&#*+!
"#H%!
"#I%!
!:!'<;)<('-!
>!+'!,)+)!=4'!./!"#F%!-'!*'+4('!/5!
!!!
!!!J)<-1+'*/,)-!=4'5!
"@K%!
!
581
!
!!!!"#$%&!'()*+,+-.!
/0+1!
!!
!!!2$3'(*&*4$!/051!6!/0+1!$'7)*)3$8.!
/091!
!!
!!!!:;87(7;(3$8!/091!)*!/<=1.!
/0>1
!!!
!!!"?@(A&*4$!)@!7)$%)3&!4)!B&;88C!8)!$'7()*)!
D!"@!(*7)E%&%!8$'%)!
C!)8!
A$3$! 8(! )F7)*4(G%&3$8! @&8! A&%E&8! H(*(7&8! 8$'%)! ;*&! 8;?)%H(A()! (*H(*(7&C! %&IJ*! ?$%! @&! A;&@! )87&!
(*7)E%&@!4)!8;?)%H(A()!%)8;@7&!8)%!A)%$.!
/0K1!
!
!!!2$3'(*&*4$!/0>1!6!/0K1!$'7)*)3$8.!
/0<1!
!!!!!L&4$!M;)!
!!!
C!@&!/0<1!8)!7%&*8H$%3&!)*.!
/001!
!!!
!!!!!!!"#$%&!'()*C!!"#$%&'()*(+,C!@&!H;)%I&!)@)A7%$87N7(A&!)*7%)!4$8!A&%E&8C!
/0,1!
!6! .!
!!
!!!L)!)87&!)A;&A(J*!)F7%&)3$8!)@!A&3?$!)@)A7%$87N7(A$!4)!@&!A&%E&! .!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
+,+-
!OP.!)*!Q;%%&6!:?()E)@C!-%.*/!/!&0"12(3/4*!/+==-1.!05C!)A;&A(J*!>'C!8)!$'7()*)!
@&!%&IJ*!4)!)87&!!4(H)%)*A(&!4)!8(E*$!A$*!%)8?)A7$!&!/051!)8!M;)!
!6!
R!
!
582
!
"#$%!
!!
!!!&'!!"#"$%&'(%)*+,&%,-./0,%,1#%,2'+3$'415'6#,2",7%$3851&%+,2%2%,7*$,
01!),*1/'2)!*()3,+'10!41+1!(5(/*,3!-(!/,641+1/'2)%!,7*()(6,389:9;!
"#;%!
!"(!')*+,-./'()-,!
!!!!!
!!<,641+1)-,!"#;%!=!"##%!,7*()(6,38!
":>%!
!?@A@B@!
!!!<,)!713(!()!"##%!C(6,3!-(+'D1-,!1#%,7%$3"!-(0!*()3,+!-(0!/164,!(0(/*+,61E)F*'/,!"(/.1/'2)!
9%!-(!01!(/.1/'2)!-(!/,)*').'-1-!=!-(!01!0(=!-(!<,.0,67G!')-(4()-'()*(6()*(!-(!013!(/.1/',)(3!
-(!H1IJ(008!
":9%!
!
!
!!!KC,+1!510*1!-(+'D1+!
G!
!=!
@!KC,+1!7'()G!
:9%G! 510*1! 3,016()*(!
@! L(/,+-(6,3! *167'F)! "(%G! 1! 317(+G!
3,016()*(!01!41+*(!(341/'10G!1!317(+G!1!317(+G!
@!<,6,!=1!*()(6,3!
!"D(+!
@! M(/(3'*16,3!
G!-(!6,-,!N.(!01!"O:%!3(!*+1)35,+61!
/,6,!3'E.(8!
"O:%
!!!
!!!!
!
":P%!
!
!
!!!L(/,+-(6,3!.)1!D(Q!6R3!01!+(E01!C8!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
9:9;
!S(+13!*'()(!01!6'361!(/.1/'2)G!4(+,!/,)!3T67,0,3!-'5(+()*(38!
G!()!-,)-(!
U!
DF13(! V,3F! S(+13G! WS,J! *,! ,7*1')! *C(! /,D1+'1)*! 5,+6! ,5! H1IJ(00X3! (N.1*',)3! 5+,)! *C(! /,)*').'*=! (N.1*',)YG! ()8!
91$*7"%#,:*1$#%&,*;,<=>+'5+?,D,0@!O>!"P>>;%8!!$Z>!
583
!
"#$%&!"'%!
!
!
!!!()*+,+),-.*!"#$%!/0!"12%!3454!.6+/0/57!
"18%!
!
!
!!!94:.!;)/<!*,!
<!!*/!*,=)/!;)/!>4!/?)4?,@0!"##%!*/!+540*A.5-4!/07!
!!!
"1$%!
!!!!B0!/>!>C-,+/!,0:/3/0:,/0+/!:/>!+,/-3.<!
"1#%!!
<!:/!-.:.!;)/7!
!
"1D%!
!
"11%!
!
!!!E'.54!6,/0<!/>!35.:)?+.!5.+4?,.04>!:/! !/*7!
"1F%!
!
"1G%!
!
!!!E'.54! 6,/0<! >.*! ,0+/=540:.*! :/! >4*! /?)4?,.0/*! "1#%<! "1D%! H! "11%! */! ,:/0+,A,?40! ?.0! >4*!
?.-3.0/0+/*!?45+/*,404*!
!:/>!5.+4?,.04>!:/! !:/!>4!/?)4?,@0!"1G%7!
"FI%!
!!!
!!!!!!!:.0:/!!
N%!
2%!
8%!
!/*!/>!*C-6.>.!:/!J/K,LM,K,+4<!;)/<!345+,/0:.!:/!
<!*/!:/A,0/!?.-.7!
<!*,!'4H!)0!0O-/5.!345!:/!3/5-)+4?,.0/*!:/!:.*!C0:,?/*!
<!*,!'4H!)0!0O-/5.!,-345!:/!3/5-)+4?,.0/*!:/!:.*!C0:,?/*!
<!/0!+.:.*!>.*!:/-P*!?4*.*!"3.5!/Q/-3>.!?)40:.!/>!-,*-.!C0:,?/!*/!5/3,+/%!
!!!M.0*,:/54-.*!;)/7!
584
!
"#$%!
!
!!!!!!!!!!!!!!
!
!
!!!!!!!!!!!!!!
!"#&%!!
!
!!!'()*+*(+,-)!"#&%!./!"#0%!1232!-4*./.3!"#5%6!
"#5%!
!!!
!!!78-32!4+./9!.:!13+,.3!*;3,+/-!<.:!:2<-!<.3.=8-!+,1:+=2!+/*.>323!(/2!=-33+./*.!?+/+*2!)-43.!(/2!
)(1.3?+=+.!+/?+/+*29!3.)(:*2/<-!./!=.3-6!
"#@%!
!!"
!A!<+?.3./=+2:!<.!)(1.3?+=+.%!
!!!B-,4+/2/<-!"#5%!C!"#@%!-4*./.,-)6!
"#D%!
!!!
!!!E.=-3<.,-)! 28-32! :2! =-33+./*.!
9! 2)-=+2<2! 2! (/2! 123*F=(:2! G(.! ).! ,(.H.!
=-/! 42I2! H.:-=+<2<! =-/)*2/*.! ! 2! :-! :23>-! <.! :2! *32C.=*-3+2! J! '()*+*(C./<-! .)*.! H2:-3! ./! "#D%!
-4*./.,-)6!
"#K%!
!!!
!!!LH2:(2/<-!.)*.!+/*.>32:9!-4*./.,-)6!
"#M%!
!!!
!!!N-3!-*32!123*.9!.:!=2,1-!,2>/;*+=-!>./.32<-!1-3!.)*2!=-33+./*.!"
:2!:.C!.P1.3+,./*2:!<.!Q+-*R'2H23*9!.)6!$M&0!
%9!!).>O/!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
$M&0
! S.32)! *+./.! :2! ,+),2! .=(2=+T/9! 1.3-! =-/! )F,4-:-)! <+?.3./*.)6!
! C!
9! ./! <-/<.!
H;2).! U-);! S.32)9! VS-W! *-! -4*2+/! *8.! =-H23+2/*! ?-3,! -?! X2PW.::Y)! .G(2*+-/)! ?3-/! *8.!
=-/*+/(+*C!.G(2*+-/Z9!./6!!"#$%&'()*$"#('+)$,)-./01203)H-:J!50!"&00[%6!!#D$!
585
!
"##$!!
!"%&'!(&!)*+,-./0/1,!2/1/!34/!2/1,563%/7!.&11/4+!"899:$;<8=$!
!
!!>&!%/!6+?2/1/6*@4!(&!"#A$!'!"##$7!6+46%3*?+B7!C3&;!
"#D$!
!!!
!
!!!!E&1*F*6/?+B!%/B!,1&B!B*G3*&4,&B!6+?2+4&4,&B7!C3&!B&!1&6+4+6&4!&4!%/!?/,1*H!":$;!
"D9$!
!
"D:$!
!
"D8$!
!
!
!!!I+4! "A9$! '! "#D$! 6+?21+J/?+B! %/! *(&4,*(/(! (&! %/B! (+B! ?/,1*6&B! C3&! B&! 2+B,3%@! &4! %/! "A:$K!
L&?+B!&46+4,1/(+!!"#$%&'"!()!#*)+,%&#()!#-"'.%#)!)-*&%/'"0+1*2-%3#"#."&*2&#()#!"#)-4"-25+#()#
-%+*2+42("(# 6# ()# !",# )-4"-2%+),# ()# 7%4!%'8# 6# ()# 92%*/:";"&*3# 2+().)+(2)+*)')+*)# ()# !",#
)-4"-2%+),#()#<"=>)!!?!L&?+B!6+11+J+1/(+!%/!M*2@,&B*B!(&!C3&!%/!
!&4!%/!&63/6*@4!"<#$!&B!
*(N4,*6/! /! %/!
! (&! %/! "=$7! OKPK>K! I+4! +,1/B! 2/%/J1/B7! M&?+B! (&1*0/(+! %/B! &63/6*+4&B! 4+M+?+GN4&/B!(&!Q/RS&%%!(&!%/!&63/6*@4!(&!6+4,*43*(/(K!!
"=$
"<#$!
!"&63/6*+4&B!*4M+?+GN4&/B$!
!
!!!!
TM+1/! J*&47! 6+4! %/! 6+11+J+1/6*@4! (&! %/! M*2@,&B*B! B+J1&!
! &R21&B/(/! &4! %/! ":$7! 2+(&?+B!
(&?+B,1/1! (*1&6,/?&4,&! %/B! &63/6*+4&B! M+?+GN4&/B! (&! Q/RS&%%7! C3&! '/! M/J5/?+B! %+G1/(+7!
6+?2+4&4,&!2+1!6+?2+4&4,&7!&4!%/!B&66*@4!C3&!0/!(&!%/!&63/6*@4!":A$!/!%/!"8U$7!/!B/J&1;!
"A$
!
!
!!!!"#"$%&#'&(")&#*"'+$#,"'+$%)&,+#-."/#"0"1%23&'"(%"/#4&$#1.&%)+#"1.&12+("$#3"1%+)2&4"$#,"#
5&67"44#+#$.#)"8)"$"(%&129(#%"($+)2&4#"(#,+$#"1.&12+("$#%"($+)2&4"$/#&#$&:")/#4&#;<=#>#4&#;?=/##$"#
,")23&(#,"#%)"$#4">"$#&62+'@%21&$#-."#49A21&'"(%"#$+(#&(%")2+)"$#&#4&$#,"#5&67"44/#&#$&:")/#4&#
"1.&129(#,"#4&#1+($")3&129(#,"#4&#1&)A&#;4&#4">#,"#B2)1**+00=C#4&#4">#,"#D+.4+':C#>#4&#4">#,"#
E2+%FG&3&)%H##
!
!!!I&# 4"># ,"# D+.4+':# ># 4&# 4"># ,"# E2+%FG&3&)%# $+(# 1&$+$# 8&)%21.4&)"$# ,"# 4&# 4"># ,"# 4&# 0.")J&# ,"#
I+)"(%JH# K4# *"1*+# ,"# -."# 4&# 4"># ,"# 4&# 0.")J&# ,"# I+)"(%J# *&# $2,+# 1+))+:+)&,&#
"68")2'"(%&4'"(%"/#,"$,"#.(#8)2(1282+/#(+#-.2%&#-."/#%"9)21&'"(%"/#"$#.(&#4">#&62+'@%21&/#"$#
,"12)/#.(&#4">#-."#(+#$"#,",.1"#,"#(2(A.(&#+%)&#4">/#49A21&'"(%"#&(%")2+)#&#4&#,"#I+)"(%JH!!
586
!
!
!!"#$#%&'!(!)&*+,*-(),.*!).%&!"#!$#%&'()!*(!*#+!$#!,-.*-/0!+!*(!*#+!$#!1&-234('(%2!$#!*(!*#+!$#!
*(!5.#%6(!$#!7-%#)26/!(!'(0#$/!1(!',2-,#*+#34564!
7458!
!!
!!!9*!#1!)('&!:#!1(!#1#)+$&'+;+,)(/!)&%&!*&!<(=!)($2('!>-#!'#!%-#?#*/!!
%(2*A+,)&!2#*#$(:&!@&$!#'+('!)($2('/!=!1(!#)-(),.*!'#!$#:-)#!(!!
7BC8!
/!*&!<(=!)(%@&!
!71#=!:#!D&-1&%08!
!!!!!E#)&$:#%&'3!
!!
7FG8!
!7FB8!
!!!
!!!D&%0,*(*:&!7BC8!)&*!7FG8!=!7FB8/!$#'@#)+,?(%#*+#/!&0+#*#%&'3!
7BH8!!
!71#=!:#!D&-1&%0!@($(!-*(!@($+I)-1(8!
7BJ8!
!!!71#=!:#!D&-1&%0!2#*#$(1,K(:(8!
!!!
!!!L<&$(! ?#$#%&'! #1! )('&! )&*+$($,&/! >-#! 1(! @($+I)-1(! '#! %-#?#! )&*! $#'@#)+&! (1! @-*+&! :&*:#! '#!
%,:#!#1!)(%@&!%(2*A+,)&/!:#!%&:&!>-#!*&!<(=!)(%@&!#1#)+$&'+;+,)&!=!'I!<(=!)(%@&!%(2*A+,)&3!
7BF8!
!
!
!!!E#)&$:#%&'!(<&$(3!
7GG8!!
!71#=!:#!M,&+NO(?($+!@($(!-*(!@($+I)-1(8!
7GJ8!
!!71#=!:#!M,&+NO(?($+!2#*#$(1,K(:(8!
!!!
!!!!O-'+,+-=#*:&!7GG8!#*!7BF8/!&0+#*#%&'3!
7B58!!
!71#=!:#!M,&+NO(?($+!@($(!-*(!@($+I)-1(8/!
!!!!#*! :&*:#! ! #'! 1(! ?#1&),:(:! :#! 1(! @($+I)-1(! ! >-#! 2#*#$(! #1! )(%@&! %(2*A+,)&/! =! ! #'! 1(!
?#1&),:(:!:#!1(!@($+I)-1(! !'&0$#!1(!)-(1!()+P(!#1!)(%@&Q!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
4564
!9'+(!@($+#!*&!?,#*#!#R@1I),+(%#*+#!#*!1&'!($+I)-1&'!:#!S#$('!
587
!
!
!!!"#$%&!'(')*)(*+$'!,-./!01!,23/!4!$5)010+$'!,2-/6!
,23/!
!
,-./!
!!!
,2-/!
!!,704!80!9*$):;&<&%)!=010%&7*>&8&/!
!!!!
!!!"#$%&'()*+),#",-*.*(#/0,#1,#2345#6#2785#",#'9)$,+,#.*#"'.0:$;+#&*(*#,.#:*%&'#,.<:)($:'#
.*"#,:0*:$'+,"#=>#==#6#=?#1,#@*AB,..C#?0@$%80+$'6!
,AB/!
#,+#
!CDEDFD!
,3G/
!!!
!
!!F0!0')&'!8$'!0@(&@*$10'!$5)010+$'6!
,22/
!
,HBB/!!
!!
!
!
!
!!!!"#$%&'()"*)+,-),./01,2,.341,,5)6%$7)(%6,)$,8'(9%,('&":*;8%,
=>,-),?'@A)$$B,?0@$%80+$'6!
,-2/!
,AI/!
,-),$'6,)8<'8;%")6,==+,===,2,
!!!
!!
!!F0!0')&'!8$'!0@(&@*$10'!$5)010+$'6!
,HBH/!
!!
!
588
!
"#$%&!
!
!!!'()! *+(,+-./.)! +012,12()!
! )345363+15! 70.! .8! /3.9:(! ! 2.;.! ).,! .<18012(! .5! .8! /3.9:(!
>#?%%!!
,./1,212(=!
!!!!"#$%&'()#*+%,'+-$&,$("#-'+./001+2+./031+'"#+%,'+'"%&$("#-'+4-5,46,6,'+6-+%,'+-$&,$("#-'+6-+
7,89-%%:+ ;&-+ #"'+ %(<-4,#+ 6-%+ =-$="+ (#$"#>-#(-#5-+ 6-+ ;&-+ %,+ -8?4-'()#+ 54,6($("#,%+ 6-+ %,'+
-$&,$("#-'+6-+7,89-%%+.@:+@@:+@@+2+@A1+#"+5"B,#+-#+$&-#5,+-8?%C$(5,B-#5-+-%+?4(#$(?("+6-+%,+
$,&',%(6,6D+E'+(B?"45,#5-+4-B,4$,4+;&-+-'5,'+6"'+'"%&$("#-'+>-$5"4(,%-'+?&-6-#+-'$4(<(4'-+
$"B"+&#,+'"%,+-$&,$()#+5-#'"4(,%:+,+',<-4:+%,+.FG1*!
"@?&!
!!!
!
+
IV. LAS ECUACIONES DE ONDA EN FORMA VECTORIAL Y TENSORIAL
!
"#! $%&! '(%)*+! #'(%',+-#&! .#! /%01#$$! &#! 2(#.#-! .#.(',*! $%&! .+&! #'(%',+-#&! .#! +-.%! .#! $+&!
'%32+&!#$4')*,'+!5!3%6-4),'+=!+(9(!1!+(5/3501+3A5!).!+(9:,0.;1B!C51!(521!.)!05!9.+153)9(!
:1,1!.8!/,15):(,/.!2.!.5.,4D1!E!3+3#-)(3B!F5!4.5.,18=!051!(521!.)!:.,3A23+1=!.)!2.+3,=!).!,.:3/.!
.5!.8!.):1+3(!"81!8(543/02!2.!(521&!E!.5!.8!/3.9:(!".8!:.,D(2(&B!'1)!(521)!70.!5.+.)3/15!05!9.23(!
:1,1! /,15):(,/1,).! )(5! (521)! 9.+G53+1)B#?%@! '1)! 605+3(5.)! :.,3A23+1)! 9G)! ).5+3881)! )(5! 81)! 2.!
).5(! E! +().5(=! 1! )1;.,!
! E!
=! 70.=! +(9(! <.9()=! )(819.5/.!
2363.,.5!.5!051!+(5)/15/.!2.!61).!(!2.!G5408(=!1!)1;.,=!
B!'1)!605+3(5.)!2.!(521!/3.5.5!
051!8(543/02! =!1!)1;.,!81!<1,31+3A5!.5!.8!.):1+3(=!70.=!.5!.8!)3)/.91=!HIJ!).!932.!.5!9./,()!E!
05!:.,D(2(! !70.!).!932.!.5!).4052()!".5!.8!+1)(!2.!81)!605+3(5.)!2.!).5(!E!+().5(!
&B!
C51! (521! /3.5.! /19;3K5! 051! <.8(+3212! ! ".5! 9./,()! :(,! ).4052(&! E! 051! 6,.+0.5+31! ! ".8!
5L9.,(! 2.! :.,D(2()! :(,! ).4052(&B! F)/1)! :,(:3.212.)! ).! ,.81+3(515! .5/,.! )D=! :(,! +01/,(!
.+01+3(5.)!2.!2.6353+3A5B#?%M!
!
!!N! :1,/3,! 2.! 81! 605+3A5! ).5(3218=! :(2.9()! 2.,3<1,! 051! "#$%&'$! ()! *$(+! ,-.! /)$)0+1#?%OB! '1!
605+3A5!2.!(521!70.!).!90.<.!%!$%!.#*#'7%!+(5!<.8(+3212!
!.)!81!)3403.5/.P!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
#?%%
Q(-5!Q1+R)(5=!8$%&&,'%$!9$#')*+.5-%3,'&=!@,2!.2B!"#SST&P!%MUB!!V(/1!W.5.P!
!!!!
#?%@
!'1)!70.!5(!8(!5.+.)3/15=!)(5!5(X9.+G53+1)B!F8!+1,G+/.,!2.!Y(521Z!2.!81)!.+01+3(5.)!2.!+19:(!5(X9.+G53+1)=!5(!
,.:,.).5/1! 05! ,1)4(! 6D)3+(! 2.8! +19:(! (! 2.! 81! :1,/D+081=! )35(! 70.! ,.:,.).5/1! 81! :,(;1;383212! 2.! +3.,/1! +(520+/1! 2.8!
+19:(!(!2.!81!:1,/D+081=!.5!.8!+(5/.[/(!2.!+3.,/1!)3/01+3A5!.[/.,51=!51/0,18!(!.[:.,39.5/18B!
#?%M
!1&!
#?%O
\!!;&!
\!!+&!
!E!!2&!
B!
!](2.9()!+(5)/,03,!(-%!:(-',;-!#&'%$%*!.#!+-.%!3<&!6#-#*%$=!1!:1,/3,!2.8!+1)(!)39:8.!2.!81!.+01+3A5!).5(3218B!
589
!
"#$$%&!
!
!!!!!'!()!*+,!-,!.+,/,!012!()!.3-.)!/,(1034)4!!"#!"$%&'$()*!5!
"#$$6&!
!
!%! 012732+)0382! 4,93/).1-! :93.,91! ()-! ,0+)0312,-! 4,! 124)! 4,! (1-! 0).:1-! ,(;079301! '!
.)<2;7301! 4,! ()-! 0+)791! ,0+)0312,-! 4,! =)>?,((@! '! ,2! -,<+34)! 01.:91A)9,.1-! *+,! ,-7)-!
,0+)0312,-! 4,! 124)! 4,! (1-! 0).:1-! ,(;079301! '! .)<2;7301! -)73-B)0,2! ()! ,0+)0382! 4,! 124)! "#$$&C!
D,0194,.1-!:93.,91!79,-!4,!()-!0+)791!,0+)0312,-!4,!=)>?,((5!
$EFC!
!G!!$$EFC!
!!!'!!!$#EFC!
!!
!
!!!H3! 71.).1-! ,2! 0+,27)! )(<+2)-! :91:3,4)4,-! 4,(! I(<,A9)! /,07193)(@! )! -)A,9@! *+,!
!'!*+,!
@!()!,0+)0382!$$EF!-,!*+,4)!)-J5!
!
"KLM&
!
!!!H+-737+3.1-!,2!"KLM&!()-!,0+)0312,-!$EF!'!$#EF!:)9)!1A7,2,9!()!"KLN&5!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
K&! O93.,9!:)-15!"K&!).:(37+4!
@!012!:,9J141!
!!
P&! H,<+241!:)-15!Q)0,.1-!.I-!<,2,9)(!()!).:(37+4@!.+(73:(30)241!()!,0+)0382!"K&!012!+2)!012-7)27,!%5!!!!!!
!!
!!!!!!!!"P&!
M&! %Q19)!<,2,9)(3R).1-!,(!:,9J141@!.+(73:(30)241!()!B)-,!012!+2!B)0719!012-7)27,! @!4,!.141!*+,!"M&!
C!%Q19)@!,(!:,9J141!')!21!,-!
@!-321!
@!01.1!01.:91A)9,.1-!)!012732+)03825!
!
N&! %Q19)!A3,2@!-3!()!124)!73,2,!+2)!(12<37+4!
@!-,!-3<+,!*+,
@!:)9)!*+,!()!,0+)0382!"M&!-,!*+,4,!
@!!:+,-!!
01.1!-3<+,5!!!"N&!
S
!
T&! H3!()!124)!-,!.+,/,!Q)03)!()!4,9,0Q)!012!/,(1034)4! @!()!,0+)0382!-,!79)2-B19.)!4,!()!-3<+3,27,!.)2,9)5!!!!!!!!!!!!!!!!!!
!!!
!,2!4124,!!
!"T&!
!-,!(().)!,(!U!"#$%&'($'&!()*@!'!
U+%$,-$!,.)')!/-0)%*C!
V&! E-7)!,0+)0382!4,!124)!!,-@!-3!()!124)!-,!.+,/,!)!()!4,9,0Q)@!!"V%&!
.+,/,!)!()!3R*+3,94)5!"V6&!
C!
!
!,-!()!
!8@!-3!()!124)!-,!
590
!
"#$%&!
!
"#$'&!
!!!!"#$%#&'()*")+(*%)*",)#%-.+)",/#01&#+)#+()2$"(0"3!"
!
!!!()!*+,-./01!234)25)5)!53!3*4)!3.-)./01!53!+15)6!.+1!,)*!7-3143*!38),-)5)*!31!4/39:+!234)25)5+6!
.+1;2-3143!.+1!3,!:2/1./:/+!53!,)!.)-*),/5)56!,)!<39+*!31.+142)5+!)22/=)6!)!*)=32>!
"#$$&!!
!
!
!!!?1!:)24/.-,)26!#$%(*+),%3)2$"(0"3)
"#$@&!
)4) )3"%()#"1+6!,)!3.-)./01!"#$'&!*3!235-.3!)>!
!!!!"#$%#&'()*")+(*%)*",)#%-.+)",/#01&#+)3&()2$"(0"3!"
!
!!!A!.+14/1-)./01!532/8)9+*!,)!3.-)./01!53,!.)9:+!9);1B4/.+C!D3.+2539+*>!
EE?FC!
G!!!!EEE?FC
!!!!!H!!!!!EI?FC!
!
!
!!J+9)15+!31!.-314)!,)*!9/*9)*!:2+:/35)53*!53,!K,;3=2)!83.4+2/),6!,)!EI?F!*3!42)1*7+29)!!
.+9+!*/;-3>!
"#$L&
!!!
!
!
!M-*4/4-/9+*!31!,)!3.-)./01!"#$L&!,)*!3.-)./+13*!!EE?F!!H!!EEE?F6!!!H!+=43139+*!,)!"#$N&>!
"#$N&""
"!!!!"#$%#&'()*")+(*%)*",)#%-.+)-%5(/0&#+)#+()2$"(0"3!"
!
!!!!O1)! 83P! 9K*6! ,)! *+,-./01! 234)25)5)! 53! 3*4)! 3.-)./01! 53! +15)6! .+1! ,)*! 7-3143*! 38),-)5)*! 31!
4/39:+! 234)25)5+! "3,! 4/39:+! 53! ,)! .)-*)! H6! :+2! ,+! 4)14+6! .+1;2-3143! .+1! 3,! :2/1./:/+! 53! ,)!
.)-*),/5)5&6!,)!<39+*!31.+142)5+!)22/=)6!)!*)=32>!
"NL&!
!
!!!
!!!?1!:)24/.-,)26!!"#$%&'(#)'*"+$,+)'
"#$Q&!
'-' ')+#$'!+.&/!,)!3.-)./01!"#$N&!*3!235-.3!)>!
!!!!"#$%#&'()*")+(*%)*",)#%-.+)-%5(/0&#+)3&()2$"(0"3!"
591
!
!!!"#! $%$&#&%#'($! )*! +,$! )-%,-#(*)$! ./012! (! ./0324! +,! )-%,-#5*! 6)! (*6,! .78824! (9&)*)'($/:;1!
4! +(! <%)! )$! -#)=&(! >(=! 6)?#*#-#5*! .@A,$)! ,==#9,24! +(! <%)! 6)'%)$&=,! <%)! +,$!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
/:;1
!!!
!B,=,!>=(9,=!)$&(4!(?=)-)'($!>=#')=(!+,!)-%,-#5*!6)!(*6,!)$-,+,=!6)!+,!*(&,!./02!)*!?(=',!@)-&(=#,+C!
D!E,!-(==(9(=,-#5*!@,!)*!$)#$!>,$($C!
!/2!E,!)-%,-#5*!6)!(*6,!@)-&(=#,+!)$!
!!
D!!
4!)*!6(*6)!
;2!E,!$(+%-#5*!6)!)$&,!)-%,-#5*!6)9)!$)=!
F!!
F!!!
F!!!
F!!!!
F!!!!G!!!!
H2!,2!
D!7)=#?#<%A'($+(D!!
!
!!!!!!!!!!
!
F!!
92!
F!!!!
-2!I*J+(K,')*&)4!
F!!!!
62!G!
)2!$%',*6(!.,24!.92!G!.-24!(9&)*)'($C!
!
D!!!
!
L2!I*J+(K,')*&)4!
I!GO5!PC!
M2!N('9#*,*6(!.H)2!G!.L24!(9&)*)'($C!.M2!
12!I2!
!!!!P2!
-((=6)*,6,$4!!
!!!!!!!!!!!)$!!
!!!!!!!!!!
!.12
!!
F!G4!>(=!6)?#*#-#5*4!
Q
F!5!
F!$#!+,!@)+(-#6,6! 4!)*!+,!6#=)--#5*!6)!%*,!6)!+,$!&=)$!
F!>(=!+(!&,*&(4!!
D!!!R,6(!<%)!&,*&(!I!.
!
2!-('(!P!.
!2!$(*!
-#)=&,$!>(=!6)?#*#-#5*!.@)=!*(&,!34!>%*&(!9!G!*(&,!/04!>%*&(!M24!$)!$#K%)!<%)!+,$!)-%,-#(*)$!6)!(*6,!6)!+($!-,'>($!
)+A-&=#-(!G!',K*A&#-(!-%'>+)*!
!G!&,'9#A*!
4!SDTDRD!
592
!
"#$%#&'(")!*+,-.!/!*+,0.!)%1&)2%#"(!3%!"#$%#&4(!5"!'(5%!*677.!/8!9':!3'!1%(1'8!)'(!"#$%#&'(")!5"!
'(5%;!!
!
!!!<(! ")1%)! "#$%#&'(")! 5"! '(5%! 3')! #%=9')! "3>#1:&#'! ! /! =%?(>1&#'! ! )"! 5")%#'93%(! /! ! !
5"9"(5"! 5"! ! /! ! /! ! 5"9"(5"! )'3%="(1"! 5"! ;! !"#$ %&'()$ *"$ +,-(.$ "-(/,0(.1"'("$
1.("12(,0)3$"-(.-$"0&.0,)'"-$%"/1,("'$)(/.-$-)#&0,)'"-$4&"$')$-".'$#.-$/"(./*.*.-3$.$-.5"/3$
#.-$-)#&0,)'"-$6.*"#.'(.*.-7$)$6.+.'8.*.-73$4&"$,1%#,0.'$!
9$:"/)3$*"#$%&'()$
*"$ +,-(.$ *"$ #.$ ;<-,0.3$ "-(.-$ -)#&0,)'"-$ .*"#.'(.*.-$ -"$ /"0=.8.'3$ %)/4&"$ +,)#.'$ "#$ %/,'0,%,)$
1"(.;<-,0.$ *"$ #.$ 0.&-.#,*.*3$ .$ -.5"/3$ -"$ /"0=.8.$ #.$ ')0,>'$ *"$ 4&"$ "#$ ";"0()$ %/"0"*"$ "'$ "#$
(,"1%)$.$#.$0.&-.9$
$$$$
$$$$@"#':5"=')!"#!$%&'#!()*+%&,#"!-)!"#+!)./#.,%*)+!-)!0#12)""!
*-.!3%)!('AB'='?>("%)C!
!
*D.!3%)!B'='?>("%)C!
!
!!E":&F%(5'!3%!*D.!#'(!:")9"#1'!%!
*++,.
!
8!'G1"("=')C!
!
!
*+++.!
!
!
!
!
!!!H$)1&1$/"(5'! *-.! "(! *+++.8! 'G1"("=')8! "(! 2':=%! 1"()':&%38! "#! )./#.,3*! -)! %*-#! -)"! .#'4%!
)").(&%'#5*6(,.%!)*!4&)+)*.,#!-)!$/)*()+7!
*++I.!
!
!
!!!J%G"! :"=%:#%:! "! B"#B'! 5"! K$"! 3%! )'3$#&4(! 1"()':&%3! :"1%:5%5%! 5"! ")1%! "#$%#&4(! *++I.! ")! 3%!
"#$%#&4(!*LD.C!
*LD.!
!!!!
!
!!!M%!"#$%#&4(!5"!'(5%!1"()':&%3!*++I.!&(1"?:%!3%)!"#$%#&'(")!5"!'(5%!F"#1':&%3")!*+,N.!/!*+,O.8!
#'='!F":"=')!%!#'(1&($%#&4(;!!E")%::'33%(5'!"3!'9":%5':!5"!EPQ3%=G":1!"(!"3!3%5'!&RK$&":5'!5"!
593
!
"#! $%&#%'()! *++,-.! /01$)$2/3! "#3! $%&#%'/)$3! 4$! /)4#! 5$%1/6'#"$3! 7#6#! "/3! %#27/3! $"8%16'%/! 9!
2#:)81'%/;!
*++<!!!=/):#2/3!
!
!9!
!
9!6$%/64$2/3!*
*++@-
!
*++A-
!
-!9!*>?-!
!!
!
!
!
!
!!!B)!"#!C"1'2#!$%&#%'()!6$%/)/%$2/3!"#!$%&#%'()!4'D$6$)%'#"!*+?A-;!
*+?A-!
!!!!"#$%#&'()*")+(*%)*",)#%-.+)",/#01&#+)#+()2$"(0"!"
!!!
!!!E/"5#2/3!#!"#!$%&#%'()!*++<-!
*++<-!
!!
!!FG/6#!3&31'1&'2/3!
!
!9!!
!
!
*++H-!
!
!
!
!!!=6/%$4$2/3!#!%#"%&"#6!"#3!%/27/)$)1$3!')4$7$)4'$)1$3!4$!"#!$%&#%'()!#)1$6'/6;!
*++>-!
!
*++J-!
*I+-!
!
*++K-!
*I,-!
!
*I<-!
!
!!L$%/64$2/3!4$!"#!*+-!M&$!
.
!*+,?-!
!
*I+-!
*+,+-!
!
*I,-!
*+,,-!
!
*I<-!
!9!
!
594
!
!
!!!"#$%#!&'$()%#!$*+#!+,-%,(./+#!#./!'%#!,.)0./+/$+#!1+!'%!+,-%,(2/!1(3+*+/,(%'!456789!
45678!!
!!!"#$%#&'()*")+(*%),%-%)".)#%/,+)/%0(12&#+)#+()3$"(2"4!
!
!!!"/!0%*$(,-'%*:!#(!!
3-+/$+#9!
455;8!
:!'%!+,-%,(2/!1+!./1%!$+/#.*(%'!#+!*+1-,+!%!'%!+,-%,(2/!#(/!
!
!!
!
!
!!!<-+!*+0*+#+/$%!'%#!1.#!+,-%,(./+#!=+,$.*(%'+#!1+!./1%:!#(/!3-+/$+:!<-+!*+#-'$%/!#+*!#()>$*(,%#!?!
+@0'A,($%)+/$+!!(/1+0+/1(+/$+#9!
456B8!
!!!!"#$%#&'()*")+(*%),%-%)".)#%/,+)".1#2-&#+)5&()3$"(2"6!
456C8!
!!!!"#$%#&'()*")+(*%),%-%)".)#%/,+)/%0(12&#+)5&()3$"(2"4!
!!!D./,'-#(2/9! E+).#! 1+*(=%1.! '%#! +,-%,(./+#! 1+! ./1%! 1+! '%#! +,-%,(./+#! 1+! F%@G+'':! $%/$.! +/!
3.*)%!=+,$.*(%'!,.).!$+/#.*(%'H!I+*.:!.%5)"#$%#&+("5)*")+(*%)#+(2&"("()/75)&(3+-/%#&'()8$")
.%5)"#$%#&+("5)*")9%:;"..6!D./!.$*%#!0%'%J*%#:!2+*%)5+.$#&'()*").%5)"#$%#&+("5)*")9%:;"..)<
8$")5+()*"),-&/"-)+-*"(==)"5)5+.$#&'()*").%5)"#$%#&+("5)*")+(*%)<8$")5+()*")5"0$(*+)+-*"(=>)
,"-+)(+)2+*%)5+.$#&'()*").%5)"#$%#&+("5)*")+(*%)"5)5+.$#&'()*").%5)"#$%#&+("5)*")9%:;"..6))
!!!!
!!!!"/! K+/+*%':! '%#! +,-%,(./+#! 1+! ./1%! $(+/+/! .$*%#! 0*.0(+1%1+#! ?! #()+$*A%#! %1(,(./%'+#! %! '%#! 1+!
F%@G+''H!"#$%!(/3.*)%,(2/!)%$+)L$(,%!%1(,(./%'!1+!'%#!+,-%,(./+#!1+!./1%:!/.!*+0*+#+/$%!M0.*!
'.!<-+!E%#$%!E.?!#+!E%!0.1(1.!,.**.J.*%*NN!!-/%!*+%'(1%1!3A#(,%H!I.*!+O+)0'.:!'%#!+,-%,(./+#!1+!
F%@G+''! K+/+*%/! ./1%#! +'+,$*.)%K/>$(,%#! $*%/#=+*#%'+#:! 0+*.! '%#! 1+! ./1%! /.! 1+$+*)(/%/! +'!
,%*L,$+*!0+*0+/1(,-'%*!1+!'%#!./1%#!+'>,$*(,%!?!)%K/>$(,%H!P-(+*.!1+,(*:!<-+!+'!E+,E.!1+!<-+!'%#!
+,-%,(./+#! 1+! ./1%! 0+*)($+/! )%$+)L$(,%)+/$+! ./1%#! +'+,$*.)%K/>$(,%#! /.N$*%/#=+*#%'+#:! /.!
#(K/(3(,%! <-+! +#$%#! ./1%#! /.N$*%/#=+*#%'+#! *+%')+/$+! +@(#$%/! +/! '%! /%$-*%'+Q%H! "/! '%! 0*L,$(,%!
*+#.'=+).#!0*()+*.!'%#!+,-%,(./+#!1+!./1%!M<-+!*+#-'$%!)L#!3L,('NN!?!.J$+/+).#! !?! :!,-?%!
*+'%,(2/!0-+1+:!+/!+#$+!,%#.:!#+*!$*%/#=+*#%'!.!/.N$*%/#=+*#%'!?!'-+K.!,.**.J.*%).#!1.#!,.#%#:!%!
#%J+*:!+/!0*()+*!'-K%*:!<-+! !?! !#%$(#3%,+/!'%#!+,-%,(./+#!1+!F%@G+''!?:!+/!#+K-/1.!'-K%*:!
<-+!'%#!#%$(#3%K%/!#.'%)+/$+!#(!#./!,%)0.#!0+*0+/1(,-'%*+#H!
!
V. POTENCIALES ELECTROMAGNÉTICOS
R+,-+*1%9!
S"TH!
SS"TH!
SSS"TH
!!!!4U+?!1+!V%-##8!
!!!!4U+?!1+!(/1-,,(2/!1+!W%*%1%?8!
!!!!!4U+?!1+!V%-##!+/!)%K/+$(#).8!
595
!
"#$%&!
!!!!!'()*!+)!,-./0)12345)667!
!
!!!%)!
!!8)!89:;)!<;)!!
'==>7!
!!
!
!!!!)?!+@?+)!
!)8!)6!A.@B)?C936!D)CB@0936EF!.@0<;)!
F!+)!-@+@!<;)!G)-@8!
)4.0)83+@! )6! C3-.@! -3:?HB9C@! C@-@! ;?3! I;?C9J?! +)6! .@B)?C936! D)CB@0936! )6)CB0@-3:?HB9C@&!
K;8B9B;*3-@8!'L7!)?!'M7N!
!
'===7!
!
!!O)C;)0+3N!
'==M7!89!
!
)?!+@?+)! !8)!663-3!)6!A.@B)?C936!)8C3630E&!
!
!!!%)!'===7!*!'==M7!@PB)?)-@8N!
'==Q7! !
!!
!
!
!!)?! +@?+)! G)-@8! )4.0)83+@! )6! C3-.@! )6HCB09C@! C@-@! ;?3! I;?C9J?! +)! 6@8! .@B)?C936)8!
)6)CB0@-3:?HB9C@8!)8C3630!*!D)CB@0936F!;83?+@!638!)C;3C9@?)8!G@-@:H?)38!+)!2345)66!'63!""!*!63!
"""7&!
!
!!,G@03!8;8B9B;90)-@8!'==Q7!)?!'"7F!*!'==>7!*!'==Q7!)?!'"#7&!K;8B9B;*)?+@!'==Q7!)?!'"7F!@PB)?)-@8N!
'==R7!
!
O)C;)0+3N!
'==L7!
!
!
%)!'==R7!*!'==L7F!@PB)?)-@8N!
'==S7!
'==T7!
!
!
!
!!!,G@03!8;8B9B;9-@8!'==>7!*!'==Q7!)?!'"#7N!
'==U7&!
!
!!O)C@0+)-@8N!
'==X7!
!!'VW!
!
7!
596
!
!
!!!"#$%&'(')#!*++,-!.!*++/-0!#%12'2$#34!
*+56-!
!
*+5+-!
!
!
!
!!!78#9(! 8(:2$#3! ;'! !"#$%&! :#'! 2<! =&'! )2! :#$>9#%(9! ?;2! <#3! :($>#3! 3#'! &'@(9&('123! %(A#! <(!
19('3=#9$(:&B'! )2! '#9$(! ?;2! (! :#'1&';(:&B'! 32! )2=&'2C! 78#9(! )2$#319($#3! ?;2! <#3! :($>#3!
2<D:19&:#!.!$(E'D1&:#!3#'!&'@(9&('123!%(A#!<(!3&E;&2'12!19('3=#9$(:&B'!)2!'#9$(4!
*+55-!
!
*+5F-!
!
!
!!G3(')#! '() &*+!,%-() .!) /0%+(0! 23! ('H<#E#! (! <(! $2)&:&B'! )2! <(! $&3$(! :#3(! :#'! )&=292'123!
&'319;$2'1#3! )2! $2)&:&B'! *>C2C! <(! <#'E&1;)! :#'! >;<E()(3! #! $219#3-0! .0! #%@&($2'120! <(! :#3(! '#!
:($%&(C!!
!
!
!!!I;31&1;&$#3!*+55-!.!*+5F-!2'!*++6-!
!!.!!*++F-!!
*+5J-! !!!
!
!!
*+5K-!
!!
!
!
!!!78#9(!%&2'0!)()#!?;24!
*+5L-!
*++F-!
0!!>(9(!#%12'294!
!!.!
!
!
!!!I2!3&E;20!:#$>(9(')#!*+5K-!.!*++F-!?;24!
*+5M-!
!
!
!!!7'H<#E($2'12!:#$>9#%($#3!?;24!
*+5,-!
!
!!!N()#!?;24!
*+5/-!
!.!
*++6-!
!0!
!
!!!32!3&E;2!?;24!
!
597
!
"#$%&!
!
!
!!!'()*+!,)-,./)0!+!-+0!.12+13)4.0!"##5&!6!-+!"#7#&8!
"##5&!
!
"##5!930&!
!
"#7#&!
!
!
!!!:0! 3/;)*<+4<.! 0.=+-+*! >2.! .0<+0! .12+13)4.0! 0)4! !"#$!%&'(!%) *!) +,%) -.,/$') !-.,-&'(!%) *!)
0,"1!++)-'2')3.(-&4()*!+)#'/!(-&,+)5!-/'$&,+)6)!%-,+,$)!+!-/$'2,7(8/&-'9!?+/.4<+@-./.4<.:)(')
!"&%/!) %'+.-&4() ,(,+;/&-,) *!) !%/,%) *'%) !-.,-&'(!%9) A34! ./@+*B)C! 03! %.#'(!2'%) +,) -'(*&-&4() *!)
<'$!(=)"4)!.0!-+!/30/+!;.*0)4+!>2.!?)*.4<D&C!>2.!0.!E.F34.!.4!-+!"#$#&C!0G!.H30<.!0)-213I48!
"#$#&!
!
!
"#$7&!
"#$$&!
!
!
!
!
!!!'()*[email protected]!12+->23.*!.12+13I4!E.!)4E+!0.!.H;*.0+!+0G8!
"#$J&!
!
!!!!.4! E)4E.! '?KL! .0! ! I! ! I!
C! E.! /)E)! >2.! -+! "#$J&! .0! 24+! .12+13I4! E.! )4E+! ;+*+! .-!
;)<.413+-!,.1<)*3+-M!!
!
!!!'()*[email protected]!!"#$%&'"&()*%#$+,-./$0$+,--/$$#)*$%&'"&()*%#$1%$)*1"2$%*$1)*1%$#3$4"0$#)!'&(5*$
6"7"$ !)#$ 7%#6%&8(9)#$ 6)8%*&("!%#$ $ 0$ :$ L@,3+/.4<.C! .0<+! <*+40F)*/+13I4! E.! -+0! *.0;.1<3,+0!
.12+13)4.0!E.;.4E.!E.!-+!,.*E+E!E.!-+!1)4E313I4!E.!?)*.4DM!N+*+!-+!1)**)@)*+13I4!E.!-+!,+-3E.D!
E.!-+!1)4E313I4!E.!?)*.4DC!(+B)!*.F.*.413+!+!O+1P0)4M#575!
!
VI. EL CARÁCTER AXIOMÁTICO DEL ELCTROMAGNETISMO
!
!!!Q+E)!>2.!-+!-.6!E.!R)2-)/@!6!-+!-.6!E.!93)<SA+,+*<!0)4!1+0)0!;+*<312-+*.0!E.!-+!-.6!E.!-+!F2.*D+!
E.! ?)*.4<DC! 0.! 03B2.! >2.! !"#$ %&'"&()*%#$ 1%$ ;"<=%!!$ #%$ 1%7(9"*2$ %*$ 7%"!(1"12$ $ 1%$ 1)#$ !%0%#$
"<()>?8(&"#2$"$#"@%72$$!"$%&'"&(5*$1%$&)*8(*'(1"1A$
$ 0$ !"$!%0$1%$!"$B'%7C"$1%$
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!
#575
!O)(4!O+1P0)4C!>+,%%&-,+)?+!-/$'*6(,2&-%C!$*E!.EM!"#TTU&87J7M!V.W!X)*PC!Y3-.6!
598
!
!"#$%&'()
*)+,&$)#$,-.&/0")#$1$./)2-$)./)$,&#-3&-#/)"#0$%/0/)4)"3-.&/)0$)./)
#$/.50/0)$,)/65"78&53/*)9"0/,)./,)7:.&5;.$,).$4$,)0$.)$.$3&#"7/<%$&5,7"=)-%/,)>?=),$)0$0-3$%)
0$)$,&/,)0",).$4$,*))
!!!!!"!#$%&'%()#'*%!)+,(-$!(%).!/0!!1#()#'$%1.!213!131#&,$4)+%1&'.4$!5(1!.1!21,'6)%!21!1.&).!2$.!
3171.!)8'$49&'#).:!
!
;),&1!<=!>?>@ABCDAEA<@"!
!F-$&1%#')!15('6)31!#$,,'1%&1!4(3&'-3'#)2)!-$,!-$&1%#')3I!(%'2)2J!K)&&L H!
FGH!
FMH!!
!!F#$,,'1%&1!1.!#),+)!-$,!(%'2)2!21!&'14-$I!(%'2)2J!"4-N,1L H!
FOH!!
!F-$&1%#')3!1.!1%1,+P)!-$,!#),+)I!(%'2)2!Q$3&.L H!
F0H!!
!F&,)R)S$!1.!-$&1%#')3!4(3&'-3'#)2)!-$,!#),+)I!(%'2)2!T$(31.L H!
FUH! ?17! 21! @$(3$4R! -),)! 2$.! #),+).! -(%&()31.! 1%! 13! 6)#P$J!
! F(%'2)2J!
V1W&$%L H!
;),&1!<<"=!@"X;C!>?Y@AB<@C!Q>@ACB<"?!!
FZH!?17!21!@$(3$4R!61#&$,')3J!
!F(%'2)2J!V1W&$%L
F[H!@)4-$!13\#&,'#$J!
F/H!@)4-$!13\#&,'#$!.'!
F]H!
!!F(%'2)21.J!
J!!
!F(%'2)21.J!
FG^H!
!#)4-$!.(-1,_'#')3J!
FGGH!
!#)4-$!&,'2'41%.'$%)3J!
FGOH!D'!
!
!F(%'2)21.J!
H!
FG0H!@)4-$!13\#&,'#$!21!(%)!#),+)!-(%&()3! !1%!1._1,)!#$%!,)2'$! !
!!!!!!!RH!61#&$,')3J!!!
!1.J!
H!
!F(%'2)21.J!
FGMH!@)4-$!13\#&,'#$!-$,!#P,#(3$!#$%!,)2'$! !1%!-3)%$!`aJ!
!!!!!!!)H!4)+%'&(2J!!
H!
!1%!2$%21!
!#)4-$!3'%1)3J!
!
!
;),&1!<<b=!c?dTC!>?Y@AB<@C!a!?>a!e>!f"dDD!
FGUH!c3(S$!13\#&,'#$J!
!!F(%'2)21.J!
H!
H!
!F(%'2)21.J!
H!
H!
!
599
!
"#$%!&'()*!+',-./0-*!+1!-0'012/*!-*1!/320*! 4!'3/5*! 6!
!"(10232+76!
%!
"#8%!93:;*!+',-./0-*!+1!;(1.*!<!5+1+/32*!;*/!='()*!+',-./0-*!+1!-0'012/*!-*1!/320*! 4!'3/5*! 6!
!!!!!!!3%!:3510.(26!!
!
!!!!!!!>%!?+-.*/03'6!!!
!
"#@%!&'()*!+',-./0-*!+1!;'31*!01=010.*6!
!"(10232+76!
%!
6!
"#A%!93:;*!+',-./0-*!5+1+/32*!;*/!='()*!+1!;'31*!01=010.*!-*1!2+170232!2+!-3/53!
!!!!!!!3%!:3510.(26!!
!
!!!!!!!>%!?+-.*/03'6!!!
!
"BC%!&'()*!+',-./0-*!+1!7(;+/=0-0+!-+//3236!!
!"(10232+76!
%!
"B#%!93:;*!+',-./0-*!5+1+/32*!;*/!+7=+/3!-*1!/320*! !D!2+170232!2+!-3/53!
!!!!!!!3%!2+1./*!2+!'3!+7=+/36!
!
!!!!!!!>%!+1!'3!7(;+/=0-0+!2+!'3!+7=+/36!!
!!!!!!!-%!=(+/3!2+!'3!+7=+/36!
!
!
!
<3/.+!EEE4!<FGHI9EJK!HKL9GME9F!
"BB%!H1+/5N3!+',-./0-36!!
!
"BO%!<*.+1-03'!+',-./0-*6!!
"BP%!93:;*!+',-./0-*!?+-.*/03'6!
!
!!!"
%!
"BQ%!93:;*!+',-./0-*!2+!(13!7*'3!-3/53! 6!!
!!!
"B$%!R*'.3)+!+1./+!2*7!;'31*7!0:35013/0*7!;3/3'+'*7!"3!207.31-03! %!3!(1!;'31*!01=010.*!-*1!!
2+170232!2+!-3/53!7(;+/=0-03'!! 6!
"B8%!R*'.3)+!+1!!7(;+/=0-0+!+7=+/3!-*1!/320*! 6!
<3/.+!ER4!9J<J9EGFMHS!T!UEHKL9GME9FS!
!";*/!#A3%!
!
600
!
"#$%!&'(')*+',)*'-!
!".,*/'/01-!
%!
!
"#2%!&'(')*+',)*'!/0!)'(')*+3401!0,!104*0-!
"56%!&'(')*+',)*'!/0!)'(')*+3401!('4'70731-!
!
"58%!&'(')*+',)*'!/0!)'(')*+34!019:4*)3!)3,!(7')'1!)3,!4'/*3!'!;!<!"'=<%-!
"5#%!>0,1*/'/!/0!0,04?@'!07:)+4*)'-!
"55%!>0,1*/'/!/0!)'4?'1!07:)+4*)'1-!
!!!".,*/'/01A!
!!".,*/'/01-!
!
%!
%!
B'4+0!CD!!EBFG&E&GHI!E!&GJ&KGLMN!
"5O%!&344*0,+0-!
!
"5P%!>0,1*/'/!/0!0,04?@'!07:)+4*)'-!
"5Q%!>0,1*/'/!/0!)'4?'1!07:)+4*)'1-!
"5R%!>0,1*/'/!/0!)344*0,+0-!
!!!".,*/'/01A!
!!".,*/'/01-!
%!
!!".,*/'/01-!
"5$%!J01*1+0,)*'-!
%!
%!
!
"52%!F0;!/0!MST-!CUJVG!!"W37+'X0!01!401*1+0,)*'!T.7+*(7*)'/'!(34!)344*0,+0Y!.,*/'/01-! %!
!
"O6%!&'T(3!07:)+4*)3-!
!
"O8%!J01*1+*W*/'/-!
"O#%!>0!"O6%!;!"O8%-!
!
!
"O5%!>0!"O#%D!F0;!/0!MST!'!,*W07!T*)431)Z(*)3-!
!
"O5%!J01*1+0,)*'!0,!('4'7073-!
!
"OO%!J01*1+0,)*'!0,!104*0-!
!
"OP%! F0;! /0! [*4)SS399! G-! 7'! 1.T'! /0! )'@/'1! /0! (3+0,)*'7! "07! /*9040,)*'7! /0! (3+0,)*'7%! 0,! .,!
)*4).*+3!)044'/3!01!)043-!
!
"OQ%F0;!/0![*4)SS399!GG-!7'!1.T'!/0!7'!)'4?'!0,+4',+0!;!1'7*0,+0!01!)043-!!
!
"OR%!F0;!/0!\3.70-!7'!(3+0,)*'!01!(43/.)+3!/0!)344*0,+0!;!W37+'X0-!
!".,*/'/01-!
%!
"O$%!J01*1+0,)*'!T.7+*(7*)'/'!)3,!)'(')*+',)*'!01!+*0T(3-!
!
B'4+0!CG-!&E]BM!]E^I_LMNL`LG&M!^_I_JE>M!BMJ!&EJ^EN!_I!]MCG]G_ILM!
"O2%!a.04b'!T'?,:+*)'-!
!".,*/'/-!L017'U %!
"P6%!]'?,*+./!9.04b'!T'?,:+*)'-!
!
"P8%!F0;!/0!7'!9.04b'!/0!F340,+b-!
!
601
!
"#$%!&'()*+!,+-./012+!345)(!+6+,5)(!24.!24))1(.0(!78!
!
"#9%!&'()*+!,+-./012+!(.!+6+,5)(:!(.!'.!;6+.4:!24.!2+,;4!,+-./0124!;();(.<12'6+)!345)(!(6!!
!!!!!!!;6+.48!
!!"(.!<4.<(!
%!
!
"#=%!>(?!<(!@140AB+C+)08!
D+)0(!E778!>FG!HF!IJDKLF!
"##%!>(?!<(!I,;M)(8!
!
"#N%!!O+,;4!,+-./0124!<(!+6+,5)(!24.!)+<14! 8!
!!!!!!!!+%!(.!6+!3';()P121(!<(6!+6+,5)(8!
!
!!!!!!!!5%!P'()+!<(6!+6+,5)(8!
!
!!!!!!!!2%!!<(.0)4!<(6!+6+,5)(8!
!
"#Q%!O+,;4!,+-./0124!(.!2(.0)4!<(!'.!346(.41<(!1<(+6!",'?!6+)-4:!31.!(P(2043!<(!54)<(%8!!
!!!!!!!
!"(.!<4.<(!
!(3!(6!.R,()4!<(!(3;1)+3!?! !6+!64.-10'<!<(6!346(.41<(%!
D+)0(!E7778!>FG!HF!&ILIHIG!
"#S%!&6'T4!,+-./01248!!
"#U%!7.<'20+.21+8!
!!"31!
!!!"'.1<+<(38!
!
%!
%!
"NV%!>(?!<(!&+)+<+?8!W&'()*+!(6(20)4,40)1*X!"P(,%8!!
"NY%!FZ;()1,(.0+6,(.0(!24.30+8!
!
!
"N$%!>(?!<(!>(.*8!6+!24))1(.0(!1.<'21<+!3(!4;4.(!+6!2+,514!<(6!2+,;4!,+-./0124!!
"N9%!F.()-[+!,+-./012+!+6,+2(.+<+8!
!
"N=%!H(.31<+<!<(!(.()-[+!,+-./012+8!
!
D+)0(!7\8!>FGFB!HF!JI\]F>>!!
!!!>+3!6(?(3!<(!J+Z^(66!(.!P4),+!1.0(-)+68!
"N#%!!
"NN%!
"NQ%!!
"NS%!
!!!">(?!<(!_+'33!(.!(6(20)121<+<%!!
!!!!">(?!<(!1.<'221`.!<(!&+)+<+?%!
!!!!!">(?!<(!_+'33!(.!,+-.(013,4%!
!!">(?!<(!I,;M)(AJ+Z^(66:!(.!<4.<(!
%!!!
602
!
!!"#$!%&'&$!(&!)#*+&%%!&,!-./0#!(1-&/&,21#%3!
4567!
!!!!!!!!!!!!!!!!!!!!!4"&'!(&!8#9$$!&,!&%&2:/121(#(7!!
4;<7!
!!!!!!!!!!!!!!4"&'!(&!1,(9221=,!(&!>#/#(#'7!
4;?7
!!!!!!!!!!!!!!!!!!!!!!!4"&'!(&!8#9$$!&,!0#@,&:1$0.7!
4;A7!
!!!4"&'!(&!B0CD/&E)#*+&%%7!!!
!!"#$!%&'&$!(&!)#*+&%%!1,(&C&,(1&,:&$!(&%!:1&0C.!4&%&2:/.$:F:12#!'!0#@,&:.$:F:12#73!
4;G7!
!!!
!
4;H7!
4;I7!
!
4;57!
!!!!
!!!"#$!%&'&$!(&!)#*+&%%!$1,!-9&,:&3!
4;;7!
!
4;J7!
4;67!
4J<7!
!
!
!
K#/:&!L3!MNOBP!Q"QRSTM)B8NUSVRBP!!
4J?7!
4JA7!!
!!!4&29#21=,!(&!.,(#!(&%!2#0C.!&%W2:/12.X!2.,!-9&,:&7!
!!4&29#21=,!(&!.,(#!(&%!2#0C.!0#@,W:12.X!2.,!-9&,:&7!
4JG7!
!!!4&29#21=,!(&!.,(#!(&%!2#0C.!&%W2:/12.X!$1,!-9&,:&7!
4JH7!!
!!4&29#21=,!(&!.,(#!(&%!2#0C.!0#@,W:12.X!$1,!-9&,:&7!
Descargar