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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).

  

Source: Anshelevich, Michael - Department of Mathematics, Texas A&M University

 

Collections: Mathematics