Vector field along a map
= Vector field along a map
{title2=$Y\in\Gamma(f^*TN)$}
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.