Let \((x^i)\) be a set of coordinate maps corresponding to a local chart \((\Omega, \varphi)\). Then \(\omega\in T^*_pM\) can be written as
\begin{equation*} \omega=\omega_idx^i|_p, \end{equation*}where \(dx^i\) denotes the differential of the coordinate map and
\begin{equation*} \omega_i=\omega(\frac{\partial }{\partial x^i}|_p). \end{equation*}