Page 265 - Schaum's Outline of Theory and Problems of Advanced Calculus

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

@ @ @
(a) Necessity. If A 1 dx þ A 2 dy þ A 3 dz ¼ d ¼ dx þ dy þ dz,then
@x @y @z
@ @ @
¼ A 1 ¼ A 2 ; ¼ A 3
ð1Þ ð2Þ ð3Þ
@x @y @z
Then by differentiating we have, assuming continuity of the partial derivatives,
@A 1 @A 2 @A 2 @A 3 @A 1 @A 3
; ;
@y ¼ @x @z ¼ @y @z ¼ @x
which is precisely the condition r A ¼ 0.
Another method: If A 1 dx þ A 2 dy þ A 3 dz ¼ d ,then
@ @ @
k ¼r
@x @y @z
A ¼ A 1 i þ A 2 j þ A 3 k ¼ i þ j þ
from which r A ¼r r ¼ 0.
Sufficiency. If r A ¼ 0,thenbyProblem 10.29, A ¼r and
@ @ @
dz ¼ d
A 1 dx þ A 2 dy þ A 3 dz ¼ A dr ¼r dr ¼ dx þ dy þ
@x @y @z
ð
ðx;y;zÞ
A 1 dx þ A 2 dy þ A 3 dz.
(b)From part (a), ðx; y; zÞ¼
ða;b;cÞ
Then omitting the integrand A 1 dx þ A 2 dy þ A 3 dz,we have
ð ð ð
ðx 2 ;y 2 ;z 2 Þ ðx 2 ;y 2 ;z 2 Þ ðx 1 ;y 1 ;z 1 Þ
¼ ¼ ðx 2 ; y 2 ; z 2 Þ ðx 1 ; y 1 ; z 1 Þ
ðx 1 ;y 1 ;z 1 Þ ða;b;cÞ ða;b;cÞ

3
2 2
2
10.31. (a)Prove that F ¼ð2xz þ 6yÞi þð6x 2yzÞj þð3x z y Þk is a conservative force field.
ð
(b) Evaluate F dr where C is any path from ð1; 1; 1Þ to ð2; 1; 1Þ. (c) Give a physical
C
interpretation of the results.
ð
(a)A force field F is conservative if the line integral F dr is independent of the path C joining any two
C
points. A necessary and sufficient condition that F be conservative is that r F ¼ 0.

i j k
@ @ @

¼ 0; F is conservative
@x @y @z
Since here r F ¼

3 2 2
2xz þ 6y 6x 2yz 3x z y
2
2
3
2 2
(b) Method 1: By Problem 10.30, F dr ¼ð2xz þ 6yÞ dx þð6x 2yzÞ dy þð3x z y Þ dz is an exact dif-
ferential d , where is such that
@ 3 @ @ 2 2 2
¼ 2xz þ 6y ¼ 6x 2yz ¼ 3x z y
ð1Þ ð2Þ ð3Þ
@x @y @z
From these we obtain, respectively,
2 3 2 2 3 2
¼ x z þ 6xy þ f 1 ð y; zÞ ¼ 6xy y z þ f 2 ðx; zÞ ¼ x z y z þ f 3 ðx; yÞ
2 3
2
These are consistent if f 1 ðy; zÞ¼ y z þ c; f 2 ðx; zÞ¼ x z þ c; f 3 ðx; yÞ¼ 6xy þ c,inwhich case
2
2 3
¼ x z þ 6xy y z þ c. Thus, by Problem 10.30,
ð
ð2;1; 1Þ
2 3 2 ð2;1; 1Þ
F dr ¼ x z þ 6xy y z þ cj ¼ 15
ð1; 1;1Þ
ð1; 1;1Þ