View source for Rotation of a vector field

{{TEX|done}}
''on a plane''

One of the characteristics of a [[Vector field|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|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|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|Degree of a mapping]]). It is related to the [[Euler characteristic|Euler characteristic]]. See also [[Poincaré theorem|Poincaré theorem]]; [[Kronecker formula|Kronecker formula]].



====Comments====
Cf. also [[Rotation number|Rotation number]] of a curve, which is the rotation of the unit tangent vector field of the curve along that curve.

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