differential geometry w notes from teacher_Part_56

differential geometry w notes from teacher_Part_56 - X is...

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5.1. LIE DERIVATIVE OF A VECTOR FIELD 111 2. Use Taylor expansion in local coordinates. ± Corollary 5.1.1 Let M be a manifold and W be a submanifold of M. Let X and Y be vector fields on M tangent to W. Then the Lie bracket [ X , Y ] is also tangent to W. di geom.tex; April 12, 2006; 17:59; p. 111
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112 CHAPTER 5. LIE DERIVATIVE 5.2 Lie Derivative of Forms and Tensors Let X be a vector field on a manifold M . Let ϕ t : M M be the flow generated by X and ϕ * t : T ϕ t ( x ) M T x M be the corresponding pullback. Definition 5.2.1 Let f be a function ( 0 -form) on M. Then the Lie derivative of f with respect to X is a function L X f defined by ( L X f ) x = d dt ( ϕ * t f ) x ± ± ± ± t = 0 = d dt f ( ϕ t ( x )) ± ± ± ± t = 0 . Proposition 5.2.1 The Lie derivative of a function f with respect to a vector field
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Unformatted text preview: X is equal to L X f = X ( f ) . In local coordinates L X f = X i i f . Denition 5.2.2 Let be a 1-form on M. The Lie derivative of with respect to X is a 1-form L X dened by ( L X ) x = lim t 1 t * t t ( x )- x = d dt ( * t ) x t = We can immediately generalize this to p-forms. Denition 5.2.3 Let be a p-form on M. The Lie derivative of with respect to X is a p-form L X dened by ( L X ) x = lim t 1 t * t t ( x )- x = d dt ( * t ) x t = di geom.tex; April 12, 2006; 17:59; p. 112...
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differential geometry w notes from teacher_Part_56 - X is...

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