In Gauge Fields, Knots, and Gravity, exercise 18 is the following: Show that if $\phi:M \to N$ we can push forward a vector field $v$ on $M$ to obtain a vector field $\phi_*$ on $N$ satisfying $(\phi_* v)_q = \phi_*(v_p)$, whenever $\phi(p)=q$.
I don't understand the question. I don't see how $(\phi_* v)_q$ is defined, given that we always apply the pushforward to tangent vectors, and not vector fields. If someone could explain the question, that would be good.
Edit: $M$ and $N$ are smooth manifolds, $\phi$ is a diffeomorphism.