Page 171 - Schaum's Outline of Theory and Problems of Advanced Calculus

162 VECTORS [CHAP. 7

Fig. 7-13 Fig. 7-14

where A 0; 0 @ < 2 ; 1 < z < 1.
Scale factors: h 1 ¼ 1; h 2 ¼ ; h 3 ¼ 1
2
2
2
2
Element of arc length: ds ¼ d þ d þ dz 2
Jacobian : @ðx; y; zÞ ¼
@ð ; ; zÞ
Element of volume: dV ¼ d d dz
Laplacian:
2
2
2
2
2
1 @ @U 1 @ U @ U @ U 1 @U 1 @ U @ U
2
@ @ @ @z @ @ @ @z
r U ¼ þ 2 2 þ 2 ¼ 2 þ þ 2 2 þ 2
Note that corresponding results can be obtained for polar coordinates in the plane by omit-
2
2
2
2
ting z dependence. In such case for example, ds ¼ d þ d , while the element of volume is
replaced by the element of area, dA ¼ d d .
2. Spherical Coordinates (r; ; Þ. See Fig. 7-14.
Transformation equations:
x ¼ r sin cos ; y ¼ r sin sin ; z ¼ r cos
where r A 0; 0 @ @ ; 0 @ < 2 .
Scale factors: h 1 ¼ 1; h 2 ¼ r; h 3 ¼ r sin
2
2
2
2
2
2
Element of arc length: ds ¼ dr þ r d þ r sin d 2
2
Jacobian : @ðx; y; zÞ ¼ r sin
@ðr; ; Þ
2
Element of volume: dV ¼ r sin dr d d
Laplacian:
2
1 @ @U 1 @ @U 1 @ U
2 2
r U ¼ 2 r þ 2 sin þ 2 2 2
r @r @r r sin @ @ r sin @
Other types of coordinate systems are possible.