# push forward of a vector field

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 $f\in C^{\infty}(N)$

• I don't understand the middle term in that equality, because I don't know what $X_\star$ means. The last one, however, looks OK. – Giuseppe Negro May 29 '18 at 8:58
• my bad. It should be $X_{\phi^{-1}(x)}$ – tomak May 29 '18 at 9:07
• Yeah, that's right. – Praneet Srivastava May 29 '18 at 9:32
• Possible duplicate of push forward of vector field – Praneet Srivastava May 29 '18 at 9:33