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Difference between revisions of "Rotation of a vector field"

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''on a plane''
 
''on a plane''
  
One of the characteristics of a [[Vector field|vector field]] that are invariant under homotopy. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826601.png" /> be a vector field on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826602.png" /> of the Euclidean plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826603.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826604.png" /> be the angle between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826605.png" /> and some fixed direction; the rotation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826606.png" /> will then be the increment of the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826607.png" /> when going around a closed oriented curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826608.png" /> along which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r0826609.png" />, divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266010.png" />. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266011.png" /> is a smooth curve of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266012.png" />, the rotation of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266013.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266014.png" />) tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266015.png" /> (or normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266016.png" />) along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266017.png" /> is equal to the total [[Curvature|curvature]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266018.png" /> divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266019.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266020.png" /> is a vector field (with or without isolated singular points) on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266021.png" />, with Jordan boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266022.png" />, then the rotation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266024.png" /> is equal to the sum of the indices of the singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266025.png" /> in the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266026.png" /> (cf. [[Singular point, index of a|Singular point, index of a]]). The rotation of a vector field remains unchanged during a homotopic deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266027.png" /> which does not pass through the singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266028.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266029.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266030.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266031.png" />, at an isolated point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266033.png" />. It is defined as the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266034.png" />, as a mapping from a small sphere around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082660/r08266035.png" /> 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]].
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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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article