
#1
Aug2210, 11:14 PM

P: 304

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? 



#2
Aug2310, 12:34 AM

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P: 4,768

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.




#3
Aug2410, 03:10 AM

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P: 9,018

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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