A smooth covector field \(\omega\) is called conservative if line integrals are path independent. That is, for two given curve segments \(\gamma\) and \(\widetilde{\gamma}\) with the same starting and end point we have
\begin{equation*} \int_{\gamma} \omega=\int_{\widetilde{\gamma}} \omega. \end{equation*}
Remark
- It is called conservative because for moving particles along a force field the energy should be conserved [@lee2013smooth_manifolds, p.292].