Vector field along a map ($Y\in \Gamma (f^{\ast }TN)$)

A vector field along a smooth map between manifolds $f:M\to N$ assigns $Y_p\in T_{f(p)}N$ smoothly to each $p$. It is a section of the pullback tangent bundle, not necessarily a vector field on $N$. Along a curve $\gamma$, the velocity $\dot{\gamma }$ is such a section.