Let \(f\colon M\to \mathbb{R}\) smooth. Then the differentiation of \(f\) is defined by
\begin{equation*} df_p(v)=vf\quad v\in T_pM. \end{equation*}This defines a smooth covector field [@lee2013smooth_manifolds].
Remarks
- this coincides with the definition of the total derivative , since in this since \(df_p\colon T_pM\to T_{f(p)}\mathbb{R}\cong \mathbb{R}\).