Proof That the Sum of the Indices of Singular Points on a Surface is Equal to the Euler Characteristic

The Euler characteristic iswhere

V is the number of vertices

E is the number of edges

F is the number of faces.

Suppose there is a map drawn on some surface.

We can change the map nto a flow so that all the vertices become sources and every face becomes a sink.

Sinks and sources both have index 1 and crosspoints have index -1 so we have

sources of index 1

crosspoints of index -1 andsinks of index 1.

Hence the sum of the indices is