Closed integration of exact form

[Jhenrique]

If $\omega$ is an exact form $( \omega = d\eta )$ and $\Omega$ is the region of integration and $\partial \Omega$ represents the boundary of integration, so the following equation is correct:
$$\\ \oint_{\partial \Omega} \omega = 0$$?

 

[Matterwave]

By Stoke's theorem:

$$\oint_{\partial\Omega} \omega =\int_{\Omega} d\omega=\int_{\Omega} dd\eta=0$$