Page 183 - Schaum's Outline of Theory and Problems of Advanced Calculus

174 VECTORS [CHAP. 7

Another method:
2
2
2
@ @ @ @ 2 @ 2 @ 2
2 2 3 2 3 2 3
r ¼ 2 þ 2 þ 2 ¼ 2 ð2x y xz Þþ 2 ð2x y xz Þþ 2 ð2x y xz Þ
@x @y @z @x @y @z
¼ 4y 6xz
7.39. Prove div curl A ¼ 0.

i j
k

div curl A ¼r ðr AÞ¼ r @=@x @=@y @=@z

A 1 A 2 A 3

@A 3 @A 2 @A 1 @A 3 @A 2 @A 1
k
@y @z @z @x @x @y
¼r i þ j þ
@ @A 3 @A 2 @ @A 1 @A 3 @ @A 2 @A 1
¼ þ þ
@x @y @z @y @z @x @z @x @y
2 2 2 2 2 2
@ A 3 @ A 2 @ A 1 @ A 3 @ A 2 @ A 1
@x @y @x @z @y @z @y @x @z @x @z @y
¼ þ þ
¼ 0
assuming that A has continuous second partial derivatives so that the order of differentiation is immaterial.
JACOBIANS AND CURVLINEAR COORDINATES
2
7.40. Find ds in (a) cylindrical and (b) spherical coordinates and determine the scale factors.
(a) Method 1:
x ¼ cos ; y ¼ sin ; ¼ z
dx ¼ sin d þ cos d ; dy ¼ cos d þ sin d ; dz ¼ dz

2
2
2
2
Then ds ¼ dx þ dy þ dz ¼ð sin d þ cos d Þ 2
2 2
þð cos d þ sin d Þ þðdzÞ
2 2 2 2 2 2 2 2 2 2
¼ðd Þ þ ðd Þ þðdzÞ ¼ h 1 ðd Þ þ h 2 ðd Þ þ d 3 ðdzÞ
and h 1 ¼ h ¼ 1, h 2 ¼ h ¼ , h 3 ¼ h z ¼ 1are the scale factors.
Method 2: The position vector is r ¼ cos i þ sin j þ zk. Then
@r @r @r
@ @ @z
dr ¼ d þ d þ dz
¼ðcos i þ sin jÞ d þð sin i þ cos jÞ d þ k dz
¼ðcos d sin d Þi þðsin d þ cos d Þj þ k dz
2 2 2 2
Thus ds ¼ dr dr ¼ðcos d sin d Þ þðsin d þ cos d Þ þðdzÞ
2 2 2 2
¼ðd Þ þ ðd Þ þðdzÞ
x ¼ r sin cos ; y ¼ r sin sin ; z ¼ r cos
ðbÞ
Then dx ¼ r sin sin d þ r cos cos d þ sin cos dr
dy ¼ r sin cos d þ r cos sin d þ sin sin dr
dz ¼ r sin d þ cos dr