Let \(\omega\in \mathfrak{X}^*(M)\) and \(dx^i\) be the coordinate covector fields on some open subset \(U\subset M\). Then

\begin{equation*} \omega=\omega_idx^i \end{equation*}

where \(\omega_i\colon U\to \mathbb{R}\) denote the components of \(\omega\) in the given chart.

Remark
  • due to (0x66c4a1cf) we have \begin{equation*} \omega_i(p)=\omega_p(\frac{\partial }{\partial x^i}{\mid}_p). \end{equation*}