Vector Operations in the Three Coordinate Systems

Vector Operations in the Three Coordinate Systems
Cartesian
a
a
a
a ------ u x ------ u y ------ u z
x
y
z
A
A
A
• A ---------x ---------y ---------z
x
y
z
A
A
A
A
A
A
A ⎛ ---------z ---------y⎞ u x ⎛ ---------x ---------z⎞ u y ⎛ ---------y ---------x⎞ u z
⎝ y
⎠
⎝
⎠
⎝
z
z
x
x
y ⎠
2
2
2
2
a -------a- -------a- -------a2
2
2
x
y
z
Cylindrical
a
1 a
a
a ------ u --- ------- u ------ u z
z
A
1 ( A )
1 A
- --- ----------- ---------z
• A --- ----------------- z
A
A
A
1 Az
1 ( A ) A ⎞
- ----------⎞ u ⎛ --------- ---------z⎞ u --- ⎛ ---------------- A ⎛ --- ---------- ---------⎠ u z
⎝ ⎝ z
⎝ z ⎠
⎠
a
⎛⎝ --------⎞⎠
2
2
'
2
1
1 a
a --- ---------------------- ----2- --------2- -------a2
z
Spherical
a
1 a
1 a
a ------ u r --- -------- u ------------- -------- u r
r r sin 2
1 ( r Ar )
1 ( A sin )
1 A
- ------------- -------------------------- ------------- ----------- • A ----2 -------------------
r
r
sin
r
sin
r
1 ( sin A ) A ⎞
A ------------- ⎛ ------------------------- ---------⎠ u r
r sin ⎝
( r A )⎞
1 ( r A ) A r⎞
1 1 Ar
- ---------------- --- ⎛ ---------- ----------⎠ u --- ⎛⎝ ---------------- ---------⎠ u ⎝
r
r sin r
r
2 a
a
⎛ r --------⎞
⎛⎝ sin --------⎞⎠
2
1
1 ⎝ r ⎠
1
a----------------------------------------------------
a ----2 ------------------------ --------------2
2
2
2
r
r
r sin r sin 2