Full text: Advanced calculus

228 
CALCULUS 
is multiply connected in both senses. If a space is linearly multiply 
connected, it is obviously multiply connected with respect to sur 
faces. For, if an arbitrarily slender tube lying in the regh i could 
be drawn together continuously to an interior point of the region 
without colliding with the boundary, the same would be true of a 
simple closed curve lying within such a tube. 
The interior of an anchor ring can be rendered linearly simply 
connected by introducing a diaphragm, as for example the cut made 
by a half-plane through the axis. It is not easy to prove that this 
is true of all the spaces bounded by a finite number of curves and 
surfaces such as are most familiar to us. So in the following we 
shall restrict ourselves to spaces that are known to have this prop 
erty. If n diaphragms are needed to render a given space linearly 
simply connected, we shall say that the original space was linearly 
(n 1 )-tuply connected. Thus the interior of an anchor ring is line 
arly doubly connected. 
7. The Integral / Pdx+ Qdy. 
Theorem 1. Let P and Q be continuous, together with dP/dy and 
dQ/dx, throughout a regionS of the plane. If the integral 
(1) 
extended along an arbitrary curve drawn in S, has the same value for 
all such curves, then 
dP = dQ 
dy dx 
(2) 
at every point of S. 
More generally, the theorem is true if the points (a, b) and (x, y), and 
the curve joining them, are restricted to lying in a square, the length of 
whose sides does not exceed a certain positive constant, li, ivliich however 
may be arbitrarily small, and ivhose centre may be any point of S. 
Let (x x , yf) be any interior point of S, which we now hold fast 
and surround by a square Si lying wholly in S. Let C be any 
simple closed curve lying in S 1} and let (a, b) and (a 1 , b') be two 
points of C, dividing C into the arcs C x and C 2 . Since by hypothesis 
Pdx + Qdy
	        
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