Page 273 - Schaum's Outline of Theory and Problems of Advanced Calculus

264 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
EG F du dv.
10.86. (a)Referring to Problem 10.85, show that the element of surface area is given by dS ¼
ðð
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
(b)Deduce from (a)that the area of a surface r ¼ rðu; vÞ is EG F du dv.
S
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

@r @r @r @r @r
@r
[Hint: Use the fact that and then use the identity
@u @v @u @v @u @v
¼
ðA BÞ ðC DÞ¼ ðA CÞðB DÞ ðA DÞðB CÞ.
10.87. (a)Prove that r ¼ a sin u cos v i þ a sin u sin v j þ a cos u,0 @ u @ ; 0 @ v < 2 represents a sphere of
2
radius a. (b)Use Problem 10.86 to show that the surface area of this sphere is 4 a .
10.88. Use the result of Problem 10.34 to obtain div A in (a)cylindrical and (b)spherical coordinates. See Page
161.