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I have the following exercise:

$M$ smooth manifold, $X$ global vector field with flow $\Phi$, $F: M \rightarrow M$ diffeomorphism. Show that the flow of $F*X$ is given by $F^{-1}(\Phi(t,F(p)))$.

Define $\varphi_t(p)=\Phi(t, F(p))$ I have to show that

$F*X(F^{-1} \circ \varphi_t \circ F)=\dfrac{\partial}{\partial t} (F^{-1} \circ \varphi_t \circ F)$ right?

So far I have:

$\dfrac{\partial}{\partial t} (F^{-1} \circ \varphi_t \circ F)=dF^{-1} \dfrac{\partial}{\partial t} \varphi_t(F(p))=dF^{-1}X(\varphi_t(F(p)))=X(F^{-1} \circ \varphi_t \circ F)$ Is that correct?

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