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*}

[1, Proposition 3.24]

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• velocity of a curve

References Link to heading

1. J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.