Indices of Vector Fields

Abstract

We want to look at continuous vector fields on an open set U in the plane, but allowing them to have a finite number of singularities. A singularity will be a point at which either the vector field is not defined, or a point where it is defined and is zero. A vector field on U with singularities in Z will therefore be a continuous mapping $$ V:U\backslash Z \to {{\mathbb{R}}^{2}}\backslash \{ 0\} , $$ where Z is a finite set in U. For vector fields arising from flow of a fluid, the singularities may arise from “sources,” where fluid is entering the system, or “sinks,” where it is leaving, or some other discontinuity.