= Index of a vector field
{title2=$\operatorname{ind}_pX$}
At an isolated zero $p$ of a smooth <vector field> $X$ on an oriented $n$-manifold, its index is the <mapping degree> of the normalized field on a sufficiently small surrounding coordinate sphere:
$$
\operatorname{ind}_pX=\deg\left(\frac{X}{\|X\|}:S^{n-1}\longrightarrow S^{n-1}\right).
$$
The sphere contains no other zeros. Shrinking it without crossing another zero gives a homotopy of normalized maps, so the degree stays unchanged.
For a nondegenerate zero, the derivative $DX_p$ is invertible and the index is $\operatorname{sign}\det DX_p$. In the plane, sources and sinks therefore have index $+1$, while a saddle has index $-1$. Degenerate zeros can have other values: $X(x,y)=(x^2-y^2,2xy)$ winds twice around a small circle and has index two.
The <Poincaré-Hopf theorem> sums these local indices to the <Euler characteristic> on a closed manifold, with the corresponding boundary condition needed when a boundary is present. Thus the local winding detects a global obstruction to a nowhere-zero field.<Manifolds, Transversality, and de Rham Cohomology by Peter Petersen>[Theorem5.4.26, Poincaré-Hopf]{ref}
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