C. Results from Analytic Function Theory
Roughly speaking, a domain is simply connected if it has no holes. More precisely, is simply connected if, for every simple closed curve in , the region enclosed by lies wholly in . For simply connected domains we have the following:
Theorem C.4 (Green's theorem) Let be a simply connected domain, and let be a piecewise-smooth simple closed curve in . Let and be functions that are continuous and that have continuous first partial derivatives in . Then
where is the region bounded by .
We first consider a simple case in which is representable in both of the forms:
One can now integrate to achieve
By a similar argument,
For more complex regions, we decompose into simple regions as above. The result then follows.
We then have the following converse to Theorem C.3.
Theorem C.5 Let and have continuous derivatives in and let be simply connected. If , then is independent of path in .
Then, by Green's Theorem (Theorem C.4),