Editing Hairy ball theorem (section)

==Counting zeros==
Every zero of a vector field has a (non-zero) "[[Vector field#Index of a vector field|index]]", and it can be shown that the sum of all of the indices at all of the zeros must be two, 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 space|compact]] [[Irregularity of a surface|regular]] 2-dimensional [[manifold]] with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.