Some further topics following Math 10B 1. In Green's theorem, we talked about integration over a closed curve C and also over its interior Summary: Some further topics following Math 10B 1. In Green's theorem, we talked about integration over a closed curve C and also over its interior D. But how do we know each curve divides the plane into the interior and exterior regions? Maybe one can construct a weird curve which gives only one region (the way the M¨obius band only has one side)? Or maybe another curve divides the plane into three regions? If you think these are ridiculous questions, take a look at the picture at the very end of the article in the next link. Does this curve have an interior and an exterior? Which of the points are inside, which outside? The fact that every simple closed curve does have an inside and an outside is important enough and hard enough to prove that is has a name, the Jordan Curve Theorem. It is taught in the Differ- ential Geometry and Topology courses. 2. A very important, and beautiful theorem in mathematics is the Cauchy Integral Formula. Let f be a complex-valued function of a complex argument z, C be a simple closed curve. Then for any point a inside C, f(a) = C f(z) z - a dz. One can prove this result using Green's Theorem. This formula is crucial in Complex Analysis (Math 165). Collections: Mathematics