Note 1 to entry: The components of an antisymmetric tensor are such that Tij=−Tji, and in particular Tii=0.
Note 2 to entry: An antisymmetric tensor defined on a three-dimensional space has three strict components which can be considered as the coordinates W1,W2,W3 of an axial vector:
(0W3−W2−W30W1W2−W10)
The axial vector associated with the antisymmetric tensor U⊗V−V⊗U is the vector product of the two vectors.
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 U⊗V−V⊗U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxkcXca WHwbGaeyOeI0IaaCOvaiabgEPielaahwfaaaa@4146@ is the vector product of the two vectors.