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Area Mathematics - General concepts and linear algebra / Scalar and vector fields

IEV ref102-05-22

en
rotation
curl
vector rot U associated at each point of a given space region with a vector U, equal to the limit of the integral over a closed surface S of the vector product of the vector surface element and the vector U, divided by the volume of the interior of the surface, when the surface is contained in a sphere the radius of which tends to zero

rotU= lim V0 1 V S e n ×UdA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGackhacaGGVbGaai iDaiaahwfacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGa amOvaiabgkziUkaaicdaaeqaaOWaaSaaaeaacaaIXaaabaGaamOvaa aadaWdwbqaaiaahwgadaWgaaWcbaGaamOBaaqabaGccqGHxdaTcaWH vbGaciizaiaadgeaaSqaaiaabofaaeqaniablkH7slabgUIiYlabgU IiYdaaaa@5061@

where endA is the vector surface element oriented outwards and V is the volume

Note 1 to entry: In orthonormal Cartesian coordinates, the three coordinates of the rotation are:

U z y U y z , U x z U z x , U y x U x y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaalaaabaGaeyOaIy RaaGPaVlaadwfadaWgaaWcbaGaamOEaaqabaaakeaacqGHciITcGaA aIPaVlaadMhaaaGaeyOeI0YaaSaaaeaacqGHciITcaaMc8Uaamyvam aaBaaaleaacaWG5baabeaaaOqaaiabgkGi2kaaykW7caWG6baaaiaa ysW7caaMe8UaaGjbVlaacYcacaaMe8UaaGjbVlaaysW7caaMe8+aaS aaaeaacqGHciITcaaMc8UaamyvamaaBaaaleaacaWG4baabeaaaOqa aiabgkGi2kaaykW7caWG6baaaiabgkHiTmaalaaabaGaeyOaIyRaaG PaVlaadwfadaWgaaWcbaGaamOEaaqabaaakeaacqGHciITcaaMc8Ua amiEaaaacaaMe8UaaGjbVlaaysW7caGGSaGaaGjbVlaaysW7caaMe8 UaaGjbVpaalaaabaGaeyOaIyRaaGPaVlaadwfadaWgaaWcbaGaamyE aaqabaaakeaacqGHciITcaaMc8UaamiEaaaacqGHsisldaWcaaqaai abgkGi2kaaykW7caWGvbWaaSbaaSqaaiaadIhaaeqaaaGcbaGaeyOa IyRaaGPaVlaadMhaaaaaaa@898E@

Note 2 to entry: The rotation of a polar vector is an axial vector and the rotation of an axial vector is a polar vector.

Note 3 to entry: The rotation of the vector field U is denoted by rotU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGacogacaGG1bGaai OCaiaacYgacaWHvbaaaa@39F6@ or ×U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaGWabiab=DGirlabgE na0kaahwfaaaa@3D4D@ . In some English texts, the rotation is denoted by curlU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGacogacaGG1bGaai OCaiaacYgacaWHvbaaaa@39F6@ .


fr
rotationnel, m
vecteur rot U associé en tout point d'un domaine déterminé de l'espace à un vecteur U, égal à la limite du quotient de l'intégrale, sur une surface fermée S, du produit vectoriel de l'élément vectoriel de surface et du vecteur U, par le volume de l'intérieur de la surface, lorsque celle-ci est contenue dans une sphère dont le rayon tend vers zéro

rotU= lim V0 1 V S e n ×UdA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGackhacaGGVbGaai iDaiaahwfacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGa amOvaiabgkziUkaaicdaaeqaaOWaaSaaaeaacaaIXaaabaGaamOvaa aadaWdwbqaaiaahwgadaWgaaWcbaGaamOBaaqabaGccqGHxdaTcaWH vbGaciizaiaadgeaaSqaaiaabofaaeqaniablkH7slabgUIiYlabgU IiYdaaaa@5061@

endA est l'élément vectoriel de surface orienté vers l'extérieur et V est le volume

Note 1 à l'article: En coordonnées cartésiennes orthonormées, les trois coordonnées du rotationnel sont

U z y U y z , U x z U z x , U y x U x y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaalaaabaGaeyOaIy RaaGPaVlaadwfadaWgaaWcbaGaamOEaaqabaaakeaacqGHciITcGaA aIPaVlaadMhaaaGaeyOeI0YaaSaaaeaacqGHciITcaaMc8Uaamyvam aaBaaaleaacaWG5baabeaaaOqaaiabgkGi2kaaykW7caWG6baaaiaa ysW7caaMe8UaaGjbVlaacYcacaaMe8UaaGjbVlaaysW7caaMe8+aaS aaaeaacqGHciITcaaMc8UaamyvamaaBaaaleaacaWG4baabeaaaOqa aiabgkGi2kaaykW7caWG6baaaiabgkHiTmaalaaabaGaeyOaIyRaaG PaVlaadwfadaWgaaWcbaGaamOEaaqabaaakeaacqGHciITcaaMc8Ua amiEaaaacaaMe8UaaGjbVlaaysW7caGGSaGaaGjbVlaaysW7caaMe8 UaaGjbVpaalaaabaGaeyOaIyRaaGPaVlaadwfadaWgaaWcbaGaamyE aaqabaaakeaacqGHciITcaaMc8UaamiEaaaacqGHsisldaWcaaqaai abgkGi2kaaykW7caWGvbWaaSbaaSqaaiaadIhaaeqaaaGcbaGaeyOa IyRaaGPaVlaadMhaaaaaaa@898E@ .

Note 2 à l'article: Le rotationnel d'un vecteur polaire est un vecteur axial et celui d'un vecteur axial est un vecteur polaire.

Note 3 à l'article: Le rotationnel du champ vectoriel U est noté rotU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGacogacaGG1bGaai OCaiaacYgacaWHvbaaaa@39F6@ ou ×U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaGWabiab=DGirlabgE na0kaahwfaaaa@3D4D@ . Dans certains textes anglais, il est noté curlU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiGacogacaGG1bGaai OCaiaacYgacaWHvbaaaa@39F6@ .


de
Rotor, m
Rotation, f

es
rotacional

ja
回転

pl
rotacja

pt
rotacional

sr
ротор, м јд

sv
rotation

zh
旋度

Publication date: 2008-08
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