 Index of a vector field

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: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 $D{X}_p$ is invertible and the index is $\mathrm{sign}\det D{X}_p$. 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.^[ref]