differential geometry w notes from teacher_Part_24

Differential - 2.4 TENSORS 47 • The collection of all contravariant tensors of rank p forms a vector space denoted by T p = E ⊗ ·· ⊗ E |{z

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Unformatted text preview: 2.4. TENSORS 47 • The collection of all contravariant tensors of rank p forms a vector space denoted by T p = E ⊗ ··· ⊗ E | {z } p . • The dimension of the vector space T p is dim T p = n p . • 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 ⊗ T of rank ( p + q ) defined by: ∀ α 1 , . . . , α p , β 1 , . . . , β q ∈ E * ( Q ⊗ T )( α 1 , . . . , α p , β 1 , . . . , β q ) = Q ( α 1 , . . . , α p ) T ( β 1 , . . . , β q ) . • The components of the tensor product Q ⊗ T are ( Q ⊗ T ) i 1 ... i p j 1 ... j q = Q i 1 ... i p T j 1 ... j q . • Thus, ⊗ : T p × T q → T p + q . • Tensor product is associative. • The basis in the space T p is e i 1 ⊗ ··· ⊗ e i p , where 1 ≤ i 1 , . . . , i p ≤ n . • 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 ⊗ ··· ⊗ 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.

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Differential - 2.4 TENSORS 47 • The collection of all contravariant tensors of rank p forms a vector space denoted by T p = E ⊗ ·· ⊗ E |{z

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