# Difference between revisions of "Rotation of a vector field"

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− | One of the characteristics of a [[Vector field|vector field]] that are invariant under homotopy. Let | + | 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 | + | 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]]. |

## Latest revision as of 18:01, 4 November 2014

*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) |

**How to Cite This Entry:**

Rotation of a vector field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Rotation_of_a_vector_field&oldid=34299