Rotation of a vector field

on a plane

One of the characteristics of a vector field that are invariant under homotopy. Let $X$ be a vector field on a domain $G$ of the Euclidean plane $E^2$ and let $\theta$ be the angle between $X$ and some fixed direction; the rotation of $X$ will then be the increment of the angle $\theta$ when going around a closed oriented curve $L\subset E^2$ along which $X\neq0$, divided by $2\pi$. For instance, if $L$ is a smooth curve of class $C^2$, the rotation of the field $\tau$ (or $\nu$) tangent to $L$ (or normal to $L$) along $L$ is equal to the total curvature of $L$ divided by $2\pi$; if $X$ is a vector field (with or without isolated singular points) on a domain $G$, with Jordan boundary $\partial G$, then the rotation of $X$ on $\partial G$ is equal to the sum of the indices of the singular points of $X$ in the closure of $G$ (cf. Singular point, index of a). The rotation of a vector field remains unchanged during a homotopic deformation of $L$ which does not pass through the singular points of $X$.

A generalization consists of the concept of the index of a vector field $v$ on an $n$-dimensional manifold $M$, at an isolated point $p$ of $v$. It is defined as the degree of $x\mapsto v(x)/|v(x)|$, as a mapping from a small sphere around $p$ to the unit sphere (cf. Degree of a mapping). It is related to the Euler characteristic. See also Poincaré theorem; Kronecker formula.

Comments

Cf. also Rotation number of a curve, which is the rotation of the unit tangent vector field of the curve along that curve.

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	W. Greub, S. Halperin, R. Vanstone, "Connections, curvature, and cohomology" , 1–3 , Acad. Press (1972)
[a3]	A. Pollack, "Differential topology" , Prentice-Hall (1974)