Plane curves, polar coordinates and winding numbers. (English) Zbl 0742.30001
The author’s 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 case, 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, and open mapping theorem in two dimensions, and assertions concerning oscillation and stability of differential equations.”.
(i) to give natural definitions of “ winding number” and “ degree” (in two dimensions); and thereby,
(ii) to deduce, with relative case, 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, and open mapping theorem in two dimensions, and assertions concerning oscillation and stability of differential equations.”.
Reviewer: J.Matkowski (Bielsko-Biała)
MSC:
| 30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 55M20 | Fixed points and coincidences in algebraic topology |
| 55M25 | Degree, winding number |
