Let \(\gamma\colon I\to M\) be a smooth curve on a manifold . Then
\begin{equation*} \gamma'(t_0):=d\gamma\Bigl(\frac{\d }{\d t}\bigg|_{t_0}\Bigr)\in T_{\gamma(t_0)}M \end{equation*}is called velocity of \(\gamma\) at \(t_0\). Note, \(d\gamma\) is the total differential of \(\gamma\) (\(\gamma\) is a smooth map between the 1-manifold \(I\) and \(M\)) and \(\d/\d t|_{t_0}\) is the usual tangent vector on the tangent space \(T_{t_0}I\). (In 1-dimensional cases we write upright \(d\) instead of \(\partial\).)