In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case of the hairy ball theorem, which simply states that there is no smooth vector field on an even-dimensional n-sphere having no sources or sinks. Formal statement Let be a differentiable manifold, of dimension , and a vector field on . Suppose that is an isolated zero of , and fix some local coordinates near . Pick a closed ball centered at , so that is the only zero of in . Then the index of at , , can be defined as the degree of the map from the boundary of to the -sphere given by . Theorem. Let be a compact differentiable manifold. Let be a vector field on with isolated zeroes. If has boundary, then we insist that be pointing in the outward normal direction along the boundary. Then we have the formula where the sum of the indices is over all the isolated zeroes of and is the Euler characteristic of . A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic 0. The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf. Significance The Euler characteristic of a closed surface is a purely topological concept, whereas the index of a vector field is purely analytic. Thus, this theorem establishes a deep link between two seemingly unrelated areas of mathematics. It is perhaps as interesting that the proof of this theorem relies heavily on integration, and, in particular, Stokes' theorem, which states that the integral of the exterior derivative of a differential form is equal to the integral of that form over the boundary. In the special case of a manifold without boundary, this amounts to saying that the integral is 0. But by examining vector fields in a sufficiently small neighborhood of a source or sink, we see that sources and sinks contribute integer amounts (known as the index) to the total, and they must all sum to 0. This result may be considered one of the earliest of a whole series of theorems establishing deep relationships between geometric and analytical or physical concepts.