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\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

The exterior derivative may be denoted in coordinate-free way. For instance, see [@lee2013smooth_manifolds, Proposition 14.32].

Here, we write down the invariant expression for a 1-form \(\omega\) evaluated on smooth vector fields \(X\) and \(Y\)

\begin{equation*} d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]), \end{equation*}

where \([X,Y]\) denotes the Lie-bracket of \(X\) and \(Y\).