|Aug22-10, 11:14 PM||#1|
defition of lie derivatives
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?
|Aug23-10, 12:34 AM||#2|
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.
|Aug24-10, 03:10 AM||#3|
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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