The Punctured Plane

I
GENERAL I ARTICLE
and (ii) there is a subdivision 0 = ao < a1 < ... < ak = 1
of [0,1] such that "( is smooth on each of the sub-intervals
Ij = [aj, aj+1]' "((0) is called the initial point and "((1) the
end point of "(. The inverse path to "( is the path "(-1 defined
by "(-l(t) = "((l-t). We shall now refer to piecewise smooth
paths simply as paths, for brevity. As before, a loop will
mean a path "( whose initial and end points are the same
point x. In this case we say the loop "( is based at x.
If "( and 7 are two paths such that the end point "((1) of "(
is the initial point 7(0) of 7, then one can form the composite
path "( * 7 defined by "( * 7(t) = "((2t) for 0 $ t $ !
and = 7(2t - 1) for! $ t $ 1. (This is the reason for introducing
piecewise smooth paths, because the composite of
smooth paths need not be a smooth path, but the composite
of piecewise smooth paths is piecewise smooth.) In particular,
we can compose two loops based at the same point.
The constant path Cx at a point x E X is defined by Cx(t) = x
for all t E [0,1]. Henceforth, we shall always assume that
X is a path connected open subset of R2, i.e.. given any two
points P and Q in X, there is a path "( in X with P as its
initial and Q as its end point.
Given a smooth vector field v = (p,q) on X, and a piecewise
smooth path "( in X, we can define the line integral
k-1
1 aj+l d"(l
d,,(2
v = ~ p("((t))-
( d + q("((t))- dt
-y j=O aj t dt
1
)
With this definition, and the standard facts about change
of variables in integration, it is easy to see that I-Y*Tv =
I-yv + IT v and I-y-l v = - I-yv. Also, for the constant path
Cx at x, we have Ie",V = o.
One is now equipped to do some algebra with (piecewise
smooth) loops. Let X be a path-connected open subset of
,42
RESONANCE I April1996