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differential geometry w notes from teacher_Part_24

differential geometry w notes from teacher_Part_24 - 47 2.4...

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2.4. TENSORS 47 The collection of all contravariant tensors of rank p forms a vector space denoted by T p = E ⊗ · · · ⊗ E 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 i 1 ,..., i p = 1 T i 1 ... i p e i 1 ⊗ · · · ⊗ e i p . The set of all tensors of type ( p , 0) forms a vector space T p of dimension n p dim T p = n p . di ff geom.tex; April 12, 2006; 17:59; p. 50
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48 CHAPTER 2. TENSORS 2.4.3 General Tensors of Type ( p , q ) A tensor of type ( p , q ) is a multi-linear real-valued functional T : E * × · · · × E * p × E × · · · × E q R The components of the tensor T with respect to the basis e i , σ i are defined
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