LINE INTEGRALS AND GREEN’S THEOREM 229 
each integral being taken from (a, b) to (a', b'), it is seen that 
J Pdx + Qdy — 0. 
C 
Hence by Theorem 2 of § 5 the relation (2) holds throughout Si and 
therefore, in particular, at (aj 1} yf) *. But the latter point was any 
interior point of S. Thus the proposition is proved in all cases. 
Theorem 2. Let P and Q be continuous, together with dP/dy and 
cQ/dx, throughout the interior of a region S of the plane, and let 
dP = dQ' 
cy ex 
If S is simply connected, the integral 
JPdx -f Qdy 
(a, ft) 
has the same value for all paths joining (a, b) with (x, y), and thus is a 
single-valued function u of (x, y). The derivatives of u exist and have 
the values 
Consider two paths, C x and C 2 , drawn in S from (a, b) to (x, y). 
If they meet only at their extremities, they form together a simple 
closed curve, C, and the integral extended along C has the value 0 
by Theorem 1 of § 5. If, however, they meet in other _ 
points, a third curve, (7 3 , can be drawn in S from (a, b) 
to (x, y) meeting each of the curves C x and C 2 only in its iJ\/Cz 
extremities. Thus the value of the integral taken over 
° Fig. 55 
C-i will be the same as when taken over C x or C 2 , and 
hence these latter values will be the same. 
It is seen, then, that the integral defines a single-valued function, 
u, throughout S. To differentiate u, let (x 0 , y 0 ) be an arbitrary 
interior point of S. Hold y fast and give to x an increment, Ax. 
The corresponding increment in u has the value 
* We are using here a slight generalization of Theorem 2, which consists in 
restricting the regions S to being simply connected. The proof holds good for 
this more general case. At the time the theorem was stated, simply connected 
regions had not been introduced.