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Area Mathematics - General concepts and linear algebra / Vectors and tensors

IEV ref102-03-43

en
antisymmetric tensor
tensor of the second order defined by a bilinear form such that f(U,V)=f(V,U) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaabYcacaaMe8UaaCOvaiaacMcacqGH9aqpcqGHsislcaWGMbGa aiikaiaahAfacaqGSaGaaGjbVlaahwfacaGGPaaaaa@473A@

Note 1 to entry: The components of an antisymmetric tensor are such that T ij = T ji MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadsfadaWgaaWcba GaamyAaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaamivamaaBaaaleaa caWGQbGaamyAaaqabaaaaa@408E@ , and in particular T ii =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadsfadaWgaaWcba GaamyAaiaadMgaaeqaaOGaeyypa0tcLbsacaaIWaaaaa@3E07@ .

Note 2 to entry: An antisymmetric tensor defined on a three-dimensional space has three strict components which can be considered as the coordinates W 1 , W 2 , W 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadEfadaWgaaWcba qcLboacaaIXaaaleqaaOGaaeilaiaabccacaWGxbWaaSbaaSqaaKqz GdGaaGOmaaWcbeaakiaabYcacaqGGaGaam4vamaaBaaaleaajug4ai aaiodaaSqabaaaaa@44DC@ of an axial vector:

( 0 W 3 W 2 W 3 0 W 1 W 2 W 1 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaabmaabaqbaeqabm WaaaqaaiaaicdaaeaacaWGxbWaaSbaaSqaaiaaiodaaeqaaaGcbaGa eyOeI0Iaam4vamaaBaaaleaacaaIYaaabeaaaOqaaiabgkHiTiaadE fadaWgaaWcbaGaaG4maaqabaaakeaacaaIWaaabaGaam4vamaaBaaa leaacaaIXaaabeaaaOqaaiaadEfadaWgaaWcbaGaaGOmaaqabaaake aacqGHsislcaWGxbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaGimaaaa aiaawIcacaGLPaaaaaa@4A36@

The axial vector associated with the antisymmetric tensor UVVU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxkcXca WHwbGaeyOeI0IaaCOvaiabgEPielaahwfaaaa@4146@ is the vector product of the two vectors.


fr
tenseur antisymétrique, m
tenseur du deuxième ordre défini par une forme bilinéaire telle que f(U,V)=f(V,U) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC yvaiaabYcacaaMe8UaaCOvaiaacMcacqGH9aqpcqGHsislcaWGMbGa aiikaiaahAfacaqGSaGaaGjbVlaahwfacaGGPaaaaa@473A@

Note 1 à l'article: Les coordonnées d'un tenseur antisymétrique sont telles que T ij = T ji MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadsfadaWgaaWcba GaamyAaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaamivamaaBaaaleaa caWGQbGaamyAaaqabaaaaa@408E@ , et en particulier T ii =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadsfadaWgaaWcba GaamyAaiaadMgaaeqaaOGaeyypa0tcLbsacaaIWaaaaa@3E07@ .

Note 2 à l'article: Un tenseur antisymétrique sur un espace à trois dimensions a trois composantes strictes qui peuvent être considérées comme les coordonnées W 1 , W 2 , W 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadEfadaWgaaWcba qcLboacaaIXaaaleqaaOGaaeilaiaabccacaWGxbWaaSbaaSqaaKqz GdGaaGOmaaWcbeaakiaabYcacaqGGaGaam4vamaaBaaaleaajug4ai aaiodaaSqabaaaaa@44DC@ d'un vecteur axial:

( 0 W 3 W 2 W 3 0 W 1 W 2 W 1 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaabmaabaqbaeqabm WaaaqaaiaaicdaaeaacaWGxbWaaSbaaSqaaiaaiodaaeqaaaGcbaGa eyOeI0Iaam4vamaaBaaaleaacaaIYaaabeaaaOqaaiabgkHiTiaadE fadaWgaaWcbaGaaG4maaqabaaakeaacaaIWaaabaGaam4vamaaBaaa leaacaaIXaaabeaaaOqaaiaadEfadaWgaaWcbaGaaGOmaaqabaaake aacqGHsislcaWGxbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaGimaaaa aiaawIcacaGLPaaaaaa@4A36@

Le vecteur axial associé au tenseur antisymétrique UVVU MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxkcXca WHwbGaeyOeI0IaaCOvaiabgEPielaahwfaaaa@4146@ est le produit vectoriel des deux vecteurs.


de
antisymmetrischer Tensor, m

es
tensor antisimétrico

ja
反対称テンソル

pl
tensor antysymetryczny
tensor skośny

pt
tensor anti-simétrico

sr
антисиметрични тензор, м јд

sv
antisymmetrisk tensor

zh
反对称张量

Publication date: 2008-08
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