Page 185 - Schaum's Outline of Theory and Problems of Advanced Calculus

176 VECTORS [CHAP. 7

where A ¼ A 1 e 1 þ A 2 e 2 þ A 3 e 3 .
1 @ ð Þð1Þ @ @ ð1Þð1Þ @ @ ð1Þð Þ @
2
ð1Þð Þð1Þ @ ð1Þ @ @ ð Þ @ @z ð1Þ @z
ðcÞ r ¼ þ þ
2
2
1 @ @ 1 @ @

@ @ @ @z
¼ þ 2 2 þ 2
MISCELLANEOUS PROBLEMS
0
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f ðrÞ 2 2 2 0
x þ y þ z and f ðrÞ¼ df =dr is assumed to exist.
r
7.43. Prove that grad f ðrÞ¼ r, where r ¼
@ @ @
f ðrÞ k
@x @y @z
grad f ðrÞ¼ r f ðrÞ¼ f ðrÞ i þ f ðrÞ j þ
@r @r @r
k
0 0 0
@x @y @z
¼ f ðrÞ i þ f ðrÞ j þ f ðrÞ
x y z 0 0
0 0 0 f ðrÞ f ðrÞ r
r r r r r
¼ f ðrÞ i þ f ðrÞ j þ f ðrÞ k ¼ ðxi þ yj þ zkÞ¼
Another method: In orthogonal curvilinear coordinates u 1 ; u 2 ; u 3 ,we have
1 @ 1 @ 1 @
e 3
r ¼ e 1 þ e 2 þ ð1Þ
h 1 @u 1 h 2 @u 2 h 3 @u 3
If, in particular, we use spherical coordinates, we have u 1 ¼ r; u 2 ¼ ; u 3 ¼ . Then letting ¼ f ðrÞ,a
function of r alone, the last two terms on the right of (1)are zero. Hence, we have, on observing that
e 1 ¼ r=r and h 1 ¼ 1, the result
1 @f ðrÞ r f ðrÞ
0
r
1 @r r r
r f ðrÞ¼ ¼ ð2Þ
7.44. (a) Find the Laplacian of ¼ f ðrÞ.(b) Prove that ¼ 1=r is a solution of Laplace’s equation
2
r ¼ 0.
(a)By Problem 7.43,
0
f ðrÞ
r
r
r ¼r f ðrÞ¼
By Problem 7.35, assuming that f ðrÞ has continuous second partial derivatives, we have

0
2 f ðrÞ
r
Laplacian of ¼r ¼r ðr Þ¼ r r

1 d
0 0 0 0
f ðrÞ f ðrÞ f ðrÞ f ðrÞ
r r r dr r r
¼r r þ ðr rÞ¼ r r þ ð3Þ
00 0 0 2
rf ðrÞ f ðrÞ 2 3 f ðrÞ
00 0
r r r
¼ 3 r þ ¼ f ðrÞþ f ðrÞ
Another method: In spherical coordinates, we have
2
1 @ 2 @U 1 @ @U 1 @ U
2
r @r @r r sin @ @ r sin @
r U ¼ 2 r þ 2 sin þ 2 2 2
If U ¼ f ðrÞ,the last two terms on the right are zero and we find
2 1 d 2 0 00 2 0
r dr r
r f ðrÞ¼ 2 ðr f ðrÞÞ ¼ f ðrÞþ f ðrÞ