I just want to check that my definition of the push forward of a vector field is correct. So I have a vector field $X$ on $M$ and a diffeomorphism $\phi:M\to N$.
Then $\phi_*X$ is defined as follows
$$(\phi_*X)(x)(f) = (\phi_{*\phi^{-1}(x)}X_{\phi^{-1}(x)})(f) = X_{\phi^{-1}(x)}(f\circ\phi)$$
where $x\in N$ and $f\in C^{\infty}(N)$