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Unformatted text preview: 2.4. TENSORS 51 2.4.5 Tensor Fields • Definition 2.4.1 A tensor field on a manifold M is a smooth assign ment of a tensor at each point of M. • Let x i α = x i α ( x β ) be a local di ff eomorphism. • Then dx i α = n X j = 1 ∂ x i α ∂ x j β dx j β and ∂ ∂ x i α = n X j = 1 ∂ x j β ∂ x i α ∂ ∂ x j β • Let T be a tensor of type ( p , q ). Then T ( α ) i 1 ... i p j 1 ... j q = n X k 1 ,..., k p = 1 n X l 1 ,..., l q = 1 ∂ x i 1 α ∂ x k 1 β ··· ∂ x i p α ∂ x k p β ∂ x l 1 β ∂ x l q α ··· ∂ x j 1 β ∂ x j q α T ( β ) k 1 ... k p l 1 ... l q 2.4.6 Tensor Bundles • Definition 2.4.2 Let M be a smooth manifold. The tensor bundle of type ( p , q ) T p q M is the collection of all tensors of type ( p , q ) at all points of M T p q M = { ( p , T )  p ∈ M , T ∈ T p q , ( x ) M } • The tensor bundle T p q M is the tensor product of the tangent and cotangent bundles T p q M = T M ⊗ ··· ⊗ T M  {z } p ⊗ T * M ⊗ ··· ⊗...
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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

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