defition of lie derivatives

wdlang

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?

quasar987

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

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.