defition of lie derivatives


by wdlang
Tags: defition, derivatives
wdlang
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#1
Aug22-10, 11:14 PM
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suppose there is a vector field V on a manifold M

V generates a flow on M

suppose \gamma(t) is an integral curce in this flow

now there is another vector field W on M

why not define the lie derivative of W with respect to V as the limit of the divide

(W(\gamma+\delta \gamma)-W(\gamma))/\delta \gamma

here the difference is taken by components

i think this is very natural from our experience in the calculus course in undergraduate.

why we need to drag?
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quasar987
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#2
Aug23-10, 12:34 AM
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The difference W(\gamma+\delta \gamma)-W(\gamma)) does not make sense since W(\gamma+\delta \gamma) and W(\gamma)) are elements of different vector spaces.
Fredrik
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#3
Aug24-10, 03:10 AM
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The expression [itex]\gamma+\delta\gamma[/itex] (which I assume means [itex]\gamma(t)+\delta\gamma(t)[/itex]) doesn't make sense either, since a manifold doesn't necessarily have a vector space structure. For most manifolds, both the + and the multiplication in [itex]\delta\gamma(t)[/itex] are undefined.


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