differential geometry w notes from teacher_Part_57

differential geometry w notes from teacher_Part_57 - 5.2....

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Unformatted text preview: 5.2. LIE DERIVATIVE OF FORMS AND TENSORS 113 Proposition 5.2.2 The Lie derivative of a p-form with respect to a vector field X is given by ( L X ) i 1 ... i p = X j j i 1 ... i p + ji 2 ... i p i 1 X j + + i 1 ... i p- 1 j i p X j Proof : 1. Use that ( * t ) i 1 ... i p ( x ) = j 1 t ( x ) x i 1 j p t ( x ) x i p j 1 ... j p ( t ( x )) In particular, for a 1-form we have ( L X ) i = X j j i + j i X j . More generally, since the flow t : M M is a di ff eomorphism it naturally acts on general tensor bundles of type ( p , q ), that is, * t : ( T p q ) t ( x ) M ( T p q ) x M For a general tensor field T of type ( p , q ) on M , * t T is a tensor field of type ( p , q ) defined by ( * t T ) k 1 ... k p i 1 ... i q ( x ) = j 1 t ( x ) x i 1 j q t ( x ) x i q x k 1 m 1 t ( x ) x k p m p t ( x ) T m 1 ... m p j 1 .......
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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 geometry w notes from teacher_Part_57 - 5.2....

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