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

Let \(M\) be a smooth manifold with or without boundary. A smooth covector field on \(M\) is exact if and only if it is conservative [@lee2013smooth_manifolds, Theorem 11.42].