Page 252 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 243

INDEPENDENCE OF THE PATH
10.11. Let Pðx; yÞ and Qðx; yÞ be continuous and have continuous first partial derivatives at each point
of a simply connected region r. Prove that a necessary and sufficient condition that
þ
Pdx þ Qdy ¼ 0 around every closed path C in r is that @P=@y ¼ @Q=@x identically in r.
C
Sufficiency. Suppose @P=@y ¼ @Q=@x. Then by Green’s theorem,
ðð
þ
@Q @P
dx dy ¼ 0
Pdx þ Qdy ¼
C @x @y
r
where r is the region bounded by C.
Necessity.
þ
Suppose Pdx þ Qdy ¼ 0around every closed path C in r and that @P=@y 6¼ @Q=@x at some point of
C
r. In particular, suppose @P=@y @Q=@x > 0atthe point ðx 0 ; y 0 Þ.
By hypothesis @P=@y and @Q=@x are continuous in r,sothat there must be some region containing
ðx 0 ; y 0 Þ as an interior point for which @P=@y @Q=@x > 0. If is the boundary of ,then by Green’s
theorem
@Q @P
þ ðð
dx dy > 0
Pdx þ Qdy ¼
@x @y

þ
contradicting the hypothesis that Pdx þ Qdy ¼ 0for all closed curves in r. Thus @Q=@x @P=@y cannot
be positive.
Similarly, we can show that @Q=@x @P=@y cannot be negative, and it follows that it must be identically
zero, i.e., @P=@y ¼ @Q=@x identically in r.
10.12. Let P and Q be defined as in Problem 10.11. Prove that a B
ð B D
necessary and sufficient condition that Pdx þ Qdy be inde-
C 1
A
pendent of the path in r joining points A and B is that A
C 2 E
@P=@y ¼ @Q=@x identically in r.
Fig. 10-12
Sufficiency. If @P=@y ¼ @Q=@x,then byProblem 10.11,
ð
Pdx þ Qdy ¼ 0
ADBEA
(see Fig. 10-12). From this, omitting for brevity the integrand Pdx þ Qdy,we have
ð ð ð ð ð ð ð
¼ 0; and so
þ ¼ ¼ ¼
C 1 C 2
ADB BEA ADB BEA AEB
i.e., the integral is independent of the path.
Necessity.
If the integral is independent of the path, then for all paths C 1 and C 2 in r we have
ð ð ð ð ð
; and ¼ 0
¼ ¼
C 1 C 2
ADB AEB ADBEA
From this it follows that the line integral around any closed path in r is zero, and hence by Problem 10.11
that @P=@y ¼ @Q=@x.