Editing Vector field (section)

===Index of a vector field===
The index of a vector field is an integer that helps describe its behaviour around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value −1 at a saddle singularity but +1 at a source or sink singularity.

Let  ''n be'' the dimension of the manifold on which the vector field is defined. Take a closed surface (homeomorphic to the (n-1)-sphere) S around the zero, so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimension ''n''&nbsp;−&nbsp;1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S<sup>''n''−1</sup>. This defines a continuous map from S to S<sup>''n''−1</sup>. The index of the vector field at the point is the [[Degree of a continuous mapping#Differential topology|degree]] of this map. It can be shown that this integer does not depend on the choice of S, and therefore depends only on the vector field itself.

The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). It is equal to +1 around a source, and more generally equal to (−1)<sup>''k''</sup> around a saddle that has ''k'' contracting dimensions and ''n''−''k'' expanding dimensions.

'''The index of the vector field''' as a whole is defined when it has just finitely many zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes.

For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the [[hairy ball theorem]].

For a vector field on a compact manifold with finitely many zeroes, the [[Poincaré-Hopf theorem]] states that the vector field’s index is the manifold’s [[Euler characteristic]].