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We define the lie derivative of a vector field [itex]Y[/itex] with respect to a vector field [itex]X[/itex] to be

[itex]L_X Y :=\operatorname{\frac{d}{dt}} |_{t=0} (\phi_t^*Y)[/itex], where [itex]\phi_t[/itex] is the flow of [itex]X[/itex]. Now how do I know this thing is differentiable? And also, I'm not sure why this thing is a vector field on [itex]M[/itex] because what if our flow is only defined on a smaller open set [itex] V \subseteq M[/itex], and then for [itex] p \in M\setminus V[/itex] surely [itex](L_X Y)_p[/itex] doesn't make sense as a vector in [itex]T_p M[/itex] ?

Anyway thank you, any help would be appreciated.