\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Let \(F\colon M\to N\) be a smooth map between two manifolds and \(\gamma\colon I\to M\) a curve on \(M\). Then the velocity of the composite curve \(F\circ \gamma\colon I\to N\) at \(t_0\in I\) is given by

\begin{equation*} (F\circ \gamma)'(t_0)=dF(\gamma'(t_0)). \end{equation*}