Hairy Ball Theorem - Counting Zeros

Counting Zeros

From a more advanced point of view: every zero of a vector field has a (non-zero) "index", and it can be shown that the sum of all of the indices at all of the zeros must be two. (This is because the Euler characteristic of the 2-sphere is two.) Therefore there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat". In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.