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Unformatted text preview: 2.4. TENSORS 47 â€¢ The collection of all contravariant tensors of rank p forms a vector space denoted by T p = E âŠ— Â·Â·Â· âŠ— E  {z } p . â€¢ The dimension of the vector space T p is dim T p = n p . â€¢ The tensor product of a contravariant tensor Q of rank p and a contravari ant tensor T of rank q is a contravariant tensor Q âŠ— T of rank ( p + q ) defined by: âˆ€ Î± 1 , . . . , Î± p , Î² 1 , . . . , Î² q âˆˆ E * ( Q âŠ— T )( Î± 1 , . . . , Î± p , Î² 1 , . . . , Î² q ) = Q ( Î± 1 , . . . , Î± p ) T ( Î² 1 , . . . , Î² q ) . â€¢ The components of the tensor product Q âŠ— T are ( Q âŠ— T ) i 1 ... i p j 1 ... j q = Q i 1 ... i p T j 1 ... j q . â€¢ Thus, âŠ— : T p Ã— T q â†’ T p + q . â€¢ Tensor product is associative. â€¢ The basis in the space T p is e i 1 âŠ— Â·Â·Â· âŠ— e i p , where 1 â‰¤ i 1 , . . . , i p â‰¤ n . â€¢ A contravariant tensor T of rank p has the form T = n X i 1 ,..., i p = 1 T i 1 ... i p e i 1 âŠ— Â·Â·Â· âŠ— e i p ....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry, Vector Space

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