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

Let \(F\colon M\to N\) smooth and \(u\colon N\to \mathbb{R}\) smooth, then

\begin{equation*} F^*(du)=d(u\circ F). \end{equation*}