Abstract
This article begins with an elementary proof of (and motivation for) the fact that a plane curve can be suitably represented in polar coordinates. Our aim is to demonstrate that this fact can be used(i) to give natural definitions of “winding number” and “degree” (in two dimensions); and thereby(ii) to deduce, with relative ease, several important, well-known results from diverse areas of mathematics including real and complex analysis, algebra, plane topology and differential equations.Among these results are the fundamental theorem of algebra, Brouwer's Fixed Point Theorem in the plane, an open mapping theorem in two dimensions, and assertions concerning oscillation and stability of differential equations.
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