# Exercise: Flows of vector fields

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?