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