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

Let \(M\) be smooth. Suppose a smooth function \(f\colon \mathbb{M}\to \mathbb{R}\) and a smooth curve segment \(\gamma\colon [a,b]\to M\). Then

\begin{equation*} \int_{\gamma} df=f(\gamma(b))-f(\gamma(b)). \end{equation*}