I have just started reading the book "A course in minimal surfaces" by Colding and Minicozzi and I have a question about the proof of the very first Lemma1.1.
We have an open domain $\Omega \subset \mathbb{R}^2$ and a two-form $\omega$ defined on $\Omega \times \mathbb{R}$. We know that $\omega$ is closed ($d\omega = 0$). Let $u: \Omega \to \mathbb{R}$ be a $C^2$ function and let $Graph(u)$ be its graph. Let $\Sigma$ be any other surface with the same boundary of $Graph(u)$.
Then in the book there is written that from Stokes' theorem we have $$ \int_{Graph(u)} \omega = \int_{\Sigma} \omega. $$
My question is: in order to apply Stokes' Lemma, we should know that $\omega$ is exact, right? We could use Poincaré's Lemma, but we don't know if $\Omega \times \mathbb{R}$ is contractible. Maybe Colding and Minicozzi are tacitly assuming that $\Omega$ is simply connected?
Thank you!