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differential geometry w notes from teacher_Part_29

differential geometry w notes from teacher_Part_29 - 57 3.1...

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3.1. EXTERIOR ALGEBRA 57 3.1.2 Permutations of Tensors Let S p be the symmetric group of order p . Then every permutation ϕ S p defines a map ϕ : T p T p , which assigns to every tensor T of type (0 , p ) a new tensor ϕ ( T ), called a permutation of the tensor T , of type (0 , p ) by: v 1 , . . . , v p ϕ ( T )( v 1 , . . . , v p ) = T ( v ϕ (1) , . . . , v ϕ ( p ) ) . Let ( i 1 , . . . , i p ) be a p -tuple of integers. Then a permutation ϕ : Z p Z p defines an action ϕ ( i 1 , . . . , i p ) = ( i ϕ (1) , . . . , i ϕ ( p ) ) . The components of the tensor ϕ ( T ) are obtained by the action of the permu- tation ϕ on the indices of the tensor T ϕ ( T ) i 1 ... i p = T i ϕ (1) ... i ϕ ( p ) . The symmetrization of the tensor T of the type (0 , p ) is defined by Sym( T ) = 1 p ! ϕ S p ϕ ( T ) . The symmetrization is also denoted by parenthesis. The components of the symmetrized tensor Sym( T ) are given by T ( i 1 ... i p ) = 1 p ! ϕ S p T i ϕ (1) ... i ϕ ( p ) . The anti-symmetrization of the tensor T of the type (0 , p ) is defined by Alt( T ) = 1 p !
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