# Equality Proof of Pushforward and Pullback

Trying to prove $df_{F(p)}(F_*(v_p)) = F^*(df_{F(p)})(v_p)$. It is given that F is a smooth function between manifolds M and N, p $\in$ M, $v_p \in T_pM$ and $df_{F(p)} \in T_{F(p)}^*N$.

• Apply the definition of $F^*$ of a 1-form $(F^*\alpha)_p(v_p) = \alpha_{F(p)}(F_*(v_p))$ for $\alpha = d f$ – Lucas Kaufmann Feb 28 '18 at 20:44
• Thank you. Just wanted to make sure I was doing it correctly. – Kris Watkins Mar 4 '18 at 18:19