Page 253 - Schaum's Outline of Theory and Problems of Advanced Calculus

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

10.13. Let P and Q be as in Problem 10.11.
(a)Prove that a necessary and sufficient condition that Pdx þ Qdy be an exact differential of a
function ðx; yÞ is that @P=@y ¼ @Q=@x.
B B
ð ð
(b) Show that in such case Pdx þ Qdy ¼ d ¼ ðBÞ ðAÞ where A and B are any two
points. A A
(a) Necessity.
@ @
dy,anexact differential, then (1) @ =@x ¼ P,(2) @ =@y ¼ 0.
If Pdx þ Qdy ¼ d ¼ dx þ
@x @y
Thus, by differentiating (1) and (2)with respect to y and x, respectively, @P=@y ¼ @Q=@x since we are
assuming continuity of the partial derivatives.
Sufficiency. ð
By Problem 10.12, if @P=@y ¼ @Q=@x,then Pdx þ Qdy is independent of the path joining two
points. In particular, let the two points be ða; bÞ and ðx; yÞ and define

ð
ðx;yÞ
Pdx þ Qdy
ðx; yÞ¼
ða;bÞ
Then
ð xþ x;y ð ðx;yÞ
Pdx þ Qdy
ðx þ x; yÞ ðx; yÞ¼ Pdx þ Qdy
ða;bÞ ða;bÞ
ð
ðxþ x;yÞ
Pdx þ Qdy
¼
ðx;yÞ
Since the last integral is independent of the path joining ðx; yÞ and ðx þ x; yÞ,wecan choose the path
to be a straight line joining these points (see Fig. 10-13) so that dy ¼ 0. Then by the mean value
theorem for integrals,
1 ð ðxþ x;yÞ
0 < < 1
ðx þ x; yÞ ðx; yÞ
x ¼ x ðx;yÞ Pdx ¼ Pðx þ x; yÞ
Taking the limit as x ! 0, we have @ =@x ¼ P.
Similarly we can show that @ =@y ¼ Q.
@ @
dy ¼ d :
Thus it follows that Pdx þ Qdy ¼ dx þ
@x @y
y

(x, y) (x + Dx, y)

(a, b)
x
Fig. 10-13