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