- #1

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[tex]\mathcal{L}_X Y(f)(p) = X(Y(f))(p) - Y(X(f)(p) = \lim_{t,s \to 0} \frac{f(\psi_s \circ \phi_t (p)) - f(\phi_t \circ \psi_s (p))}{st}[/tex]

where [tex]\phi[/tex] and [tex]\psi[/tex] are the flows generated by X and Y respectively.

On the other hand, using an alternative definitiion of the Lie derivative

[tex](\mathcal{L}_X Y)_p=\left.\frac{d}{dt}\right|_{t=0}\left((\phi_t^{-1})_*Y_{\phi_{t}(p)}\right)[/tex]

we get

[tex]\mathcal{L}_X Y(f)(p) = \lim_{t,s \to 0} \frac{f( \phi_t^{-1} \circ \psi_s \circ \phi_t (p)) - f(\psi_s (p))}{st}[/tex]

Are these equal?