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Unformatted text preview: E *  ±±±±±±±±±± {z ±±±±±±±±±± } p → R • Remarks. • The function T ( α 1 , ··· , α p ) is linear in each argument. • The functional T is independent of any basis. • A contravariant vector (covector) is a contravariant tensor of rank 1. • The components of the tensor T with respect to the basis σ i are deﬁned by T i 1 ... i p = T ( σ i 1 , . . . , σ i p ) . • Then for any covectors α ( a ) = n X j = 1 α ( a ) j σ j , where a = 1 , . . . , p , we have T ( α (1) , . . . , α ( p ) ) = n X j 1 ,..., j p = 1 T j 1 ... j p α (1) j 1 ··· α ( p ) j p . • The inverse matrix of the components of a metric tensor deﬁnes a contravariant tensor g1 of rank 2 by g1 ( α, β ) = n X i , j = 1 g i j α i β j . di ﬀ geom.tex; April 12, 2006; 17:59; p. 49...
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 Spring '10
 Wong
 Geometry, Vectors, Vector Space, Tensor, Covariance and contravariance of vectors, tensor product, tensor of type, covariant tensor

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