differential geometry w notes from teacher_Part_23

differential geometry w notes from teacher_Part_23 - E |...

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2.4. TENSORS 45 The dimension of the vector space T p is dim T p = n p . The tensor product of two covectors α, β E * is a covariant tensor α β E * E * of rank 2 defined by: v , w E ( α β )( v , w ) = α ( v ) β ( w ) . The components of the tensor product α β are ( α β ) i j = α i β j . The tensor product of a covariant tensor Q of rank p and a covariant ten- sor T of rank q is a covariant tensor Q T of rank ( p + q ) defined by: v 1 , . . . , v p , w 1 , . . . , w q E ( Q T )( v 1 , . . . , v p , w 1 , . . . , w q ) = Q ( v 1 , . . . , v p ) T ( w 1 , . . . , w 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 σ i 1 ⊗ ··· ⊗ σ i p , where 1 i 1 , . . . , i p n . A covariant tensor Q of rank p has the form Q = n X i 1 ,..., i p = 1 Q i 1 ... i p σ i 1 ⊗ ··· ⊗ σ i p . di geom.tex; April 12, 2006; 17:59; p. 48
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46 CHAPTER 2. TENSORS 2.4.2 Contravariant Tensors A contravariant vector can be considered as a linear real-valued functional v : E * R . A contravariant tensor of rank p (or a tensor of type ( p , 0)) is a multi- linear real-valued functional T : E * × ··· ×
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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 con-travariant tensor g-1 of rank 2 by g-1 ( 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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