Page 264 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 255

þ
10.28. Prove that a necessary and sufficient condition that A dr ¼ 0 for every closed curve C is that
r A ¼ 0 identically. C
Sufficiency. Suppose r A ¼ 0. Then by Stokes’ theorem
þ ðð
ðr AÞ n dS ¼ 0
A dr ¼
C
S
Necessity.
þ
Suppose A dr ¼ 0around every closed path C, and assume r A 6¼ 0 at some point P. Then
C
assuming r A is continuous, there will be a region with P as an interior point, where r A 6¼ 0.Let S be
asurface contained in this region whose normal n at each point has the same direction as r A, i.e.,
r A ¼ n where is a positive constant. Let C be the boundary of S. Then by Stokes’ theorem
þ ðð ðð
ðr AÞ n dS ¼ n n dS > 0
A dr ¼
C
S S
þ
which contradicts the hypothesis that A dr ¼ 0 and shows that r A ¼ 0.
C ð P 2
It follows that r A ¼ 0 is also a necessary and sufficient condition for a line integral A dr to be
independent of the path joining points P 1 and P 2 . P 1
10.29. Prove that a necessary and sufficient condition that r A ¼ 0 is that A ¼r .
Sufficiency. If A ¼r ,then r A ¼r r ¼ 0 by Problem 7.80, Chap. 7, Page 179.
Necessity. þ ð
If r A ¼ 0,thenby Problem 10.28, A dr ¼ 0 around every closed path and A dr is independent
C
of the path joining two points which we take as ða; b; cÞ and ðx; y; zÞ. Let us define
ð ð
ðx;y;zÞ ðx;y;zÞ
A 1 dx þ A 2 dy þ A 3 dz
ðx; y; zÞ¼ A dr ¼
ða;b;cÞ ða;b;cÞ
Then
ð
ðxþ x;y;zÞ
A 1 dx þ A 2 dy þ A 3 dz
ðx þ x; y; zÞ ðx; y; zÞ¼
ðx;y;zÞ
Since the last integral is independent of the path joining ðx; y; zÞ and ðx þ x; y; zÞ,wecan choose the
path to be a straight line joining these points so that dy and dz are zero. Then
1 ð ðxþ x;y;zÞ
0 < < 1
ðx þ x; y; zÞ ðx; y; zÞ
x ¼ x ðx;y;zÞ A 1 dx ¼ A 1 ðx þ x; y; zÞ
where we have applied the law of the mean for integrals.
Taking the limit of both sides as x ! 0gives @ =@x ¼ A 1 .
Similarly, we can show that @ =@y ¼ A 2 ; @ =@z ¼ A 3 :
@ @ @
k ¼r :
@x @y @z
Thus, A ¼ A 1 i þ A 2 j þ A 3 k ¼ i þ j þ
10.30. (a) Prove that a necessary and sufficient condition that A 1 dx þ A 2 dy þ A 3 dz ¼ d ,an exact
differential, is that r A ¼ 0 where A ¼ A 1 i þ A 2 j þ A 3 k.
(b) Show that in such case,
ð ð
ðx 2 ;y 2 ;z 2 Þ ðx 2 ;y 2 ;z 2 Þ
A 1 dx þ A 2 dy þ A 3 dz ¼ d ¼ ðx 2 ; y 2 ; z 2 Þ ðx 1 ; y 1 ; z 1 Þ
ðx 1 ;y 1 ;z 1 Þ ðx 1 ;y 1 ;z 1 Þ