Page 263 - Schaum's Outline of Theory and Problems of Advanced Calculus

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

or
ðð þ
A 1 dx
½r ðA 1 iފ n dS ¼
C
S
Similarly, by projections on the other coordinate planes,
ðð þ ðð þ
A 2 dy; A 3 dz
½r ðA 2 jފ n dS ¼ ½r ðA 3 kފ n dS ¼
C C
S S
Thus, by addition,
ðð þ
A dr
ðr AÞ n dS ¼
C
S
The theorem is also valid for surfaces S which may not satisfy the restrictions imposed above. For
assume that S can be subdivided into surfaces S 1 ; S 2 ; ... ; S k with boundaries C 1 ; C 2 ; .. . ; C k which do satisfy
the restrictions. Then Stokes’ theorem holds for each such surface. Adding these surface integrals, the total
surface integral over S is obtained. Adding the corresponding line integrals over C 1 ; C 2 ; ... ; C k ,the line
integral over C is obtained.

2
10.27. Verify Stoke’s theorem for A ¼ 3yi xzj þ yz k, where S is
2 2
the surface of the paraboloid 2z ¼ x þ y bounded by z ¼ 2
and C is its boundary. See Fig. 10-21.
The boundary C of S is a circle with equations
2
2
x þ y ¼ 4; z ¼ 2 and parametric equations x ¼ 2cos t; y ¼
2 sin t; z ¼ 2, where 0 @ t < 2 . Then
þ þ
2
3ydx xz dy þ yz dz
A dr ¼
C C
ð 0
3ð2 sin tÞð 2 sin tÞ dt ð2cos tÞð2Þð2cos tÞ dt
¼
2
ð 2
2
2
ð12 sin t þ 8cos tÞ dt ¼ 20
¼
0
Fig. 10-21

i j
k

@ @ @
2
Also, r A ¼ ¼ðz þ xÞi ðz þ 3Þk

@x @y z

3y xz yz 2
2 2 xi þ yj k
and rðx þ y 2zÞ ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
2 2 2 2
n ¼
jrðx þ y 2zÞj x þ y þ 1
Then
ðð ðð ðð
dx dy
2
2
ðr AÞ n ðxz þ x þ z þ 3Þ dx dy
ðr AÞ n dS ¼ ¼
jn kj
S r r
! 2
8 9
2
2
x þ y 2 x þ y 2 =
ðð <
x 2 þ 3 dx dy
2 2
¼ þx þ
: ;
r
In polar coordinates this becomes
ð 2 ð 2
4
2
2
2
fð cos Þð =2Þþ cos þ =2 þ 3g d d ¼ 20
¼0 ¼0