Page 271 - Schaum's Outline of Theory and Problems of Advanced Calculus

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

ðð
10.66. Prove that n dS ¼ 0, where n is the outward drawn normal to any closed surface S.[Hint: Let A ¼ c,
S
where c is an arbitrary vector constant. Express the divergence theorem in this special case. Use the
arbitrary property of c.

10.67. If n is the unit outward drawn normal to any closed surface S bounding the region V,prove that
ððð
div n dV ¼ S
V

STOKES’ THEOREM
2
10.68. Verify Stokes’ theorem for A ¼ 2yi þ 3xj z k, where S is the upper half surface of the sphere
2
2
2
x þ y þ z ¼ 9 and C is its boundary. Ans. common value ¼ 9
2
10.69. Verify Stokes’ theorem for A ¼ð y þ zÞi xzj þ y k, where S is the surface of the region in the first octant
bounded by 2x þ z ¼ 6 and y ¼ 2 which is not included in the (a) xy plane, (b)plane y ¼ 2, (c)plane
2x þ z ¼ 6 and C is the corresponding boundary.
Ans. The common value is (a) 6; ðbÞ 9; ðcÞ 18
ðð
2
3
10.70. Evaluate ðr AÞ n dS, where A ¼ðx zÞi þðx þ yzÞj 3xy k and S is the surface of the cone
S
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
x þ y above the xy plane. Ans: 12
z ¼ 2
ðð
10.71. If V is a region bounded by a closed surface S and B ¼r A,prove that B n dS ¼ 0.
S
2
10.72. (a)Prove that F ¼ð2xy þ 3Þi þðx 4zÞj 4yk is a conservative force field. (b)Find such that F ¼r .
ð
(c) Evaluate F dr, where C is any path from ð3; 1; 2Þ to ð2; 1; 1Þ.
C
2
Ans: ðbÞ ¼ x y 4yz þ 3x þ constant; (c)6
2
2
2
10.73. Let C be any path joining any point on the sphere x þ y þ z ¼ a 2 to any point on the sphere
ð
5
3
5
2
2
2
2
x þ y þ z ¼ b . Show that if F ¼ 5r r, where r ¼ xi þ yj þ zk,then F dr ¼ b a .
C
ð
10.74. In Problem 10.73 evaluate F dr is F ¼ f ðrÞr, where f ðrÞ is assumed to be continuous.
ð b C
Ans: rf ðrÞ dr
a
10.75. Determine whether there is a function such that F ¼r , where:
2
3
2
(a) F ¼ðxz yÞi þðx y þ z Þj þð3xz xyÞk:
y
2 y
(b) F ¼ 2xe i þðcos z x e Þj y sin zk. If so, find it.
2 y
Ans: ðaÞ does not exist. ðbÞ ¼ x e þ y cos z þ constant
3
2
2
10.76. Solve the differential equation ðz 4xyÞ dx þð6y 2x Þ dy þð3xz þ 1Þ dz ¼ 0.
2
3
2
Ans: xz 2x y þ 3y þ z ¼ constant
MISCELLANEOUS PROBLEMS
þ
@U @U
10.77. Prove that a necessary and sufficient condition that dy dx be zero around every simple closed
C @x @y
path C in a region r (where U is continuous and has continuous partial derivatives of order two, at least) is
2
2
@ U @ U
that þ ¼ 0.
@x 2 @y 2