Let \(\omega\) be a smooth covector field . It is closed if
\begin{equation*} \frac{\partial \omega_j}{\partial x^i}=\frac{\partial \omega_i}{\partial x^j}, \end{equation*}for all pairs \(i\) and \(j\).
Remarks
- this is motivated by Schwarz Theorem applied on a potential function.
- being close is a coordinate-independent property, i.e. it suffices to show the closeness property for one specific local chart [@lee2013smooth_manifolds, Proposition 11.45].