C. Results from Analytic Function Theory
Consider functions of two independent variables, and . (The reader can think of as the real axis and as the imaginary axis.)
Let and be two functions of and , continuous in some domain . Say we have a curve in , described by the parametric equations
We can then define the following line integrals along the path from point to point inside .
We then have the following result.
hold throughout . Conversely, if a function can be found such that (C.2.4) hold, then is independent of the path.
Suppose that the integral is independent of the path in . Then, choose a point in and let be defined as follows
where the integral is taken on an arbitrary path in joining and . Because the integral is independent of the path, the integral does indeed depend only on and defines the function . It remains to establish (C.2.4).
For a particular in , choose so that and so that the line segment from to in is as shown in Figure C.1. Because of independence of the path,
We think of and as being fixed while may vary along the horizontal line segment. Thus is being considered as function of . The first integral on the right-hand side of (C.2.6) is then independent of .
Hence, for fixed , we can write
The fundamental theorem of Calculus now gives
A similar argument shows that
Conversely, let (C.2.4) hold for some . Then, with as a parameter,
on every closed path in . Conversely if (C.2.13) holds for every simple closed path in , then is independent of the path in .
Suppose that the integral is independent of the path. Let be a simple closed path in , and divide into arcs and as in Figure C.2.
The converse result is established by reversing the above argument.
Actually, we will be particularly interested in the converse to Theorem C.3. However, this holds under slightly more restrictive assumptions, namely a simply connected domain.