Is there a reason one cannot apply Stokes theorem $\int_{\Sigma} d \omega = \int_{\partial \Sigma} \omega$, where $\partial \Sigma$ is an orientable, compact manifold with genus $g\geq 1$? For example, could I apply this relation for $\partial \Sigma$ a torus and $\Sigma$ the space contained within the torus (a donut) or the complement of that donut?
Differently stated, must $\Sigma$ be simply connected in order to apply Stokes theorem?
Thank you!
Sure you can apply Stokes theorem. Cut your torus vertically (so you have two cylinders), then apply Stokes theorem on each piece.
Edit: Doesn't have to be simply connected (recall Green's theorem).