Let M a compact manifold $C^k$ with boundary and orientable such that $\partial M \neq 0$ and $\omega \in \Lambda^{dim M -1}$ such that $d\omega (x) \neq 0, \forall x \in M$. Show that exists $x_0 \in \partial M$ such that $\omega(x_0) \neq 0$.
Hint: Start with $\overline{B(0,1)}$.
It's a homework problem.
My first idea was: $dω(x)≠0$, then exists an open set U such that $dω(U)≠0$. Let's say that $dω(U)>0⇒∫Udω>0$. Then use the Stokes theorem to conclude something like $∫_Vω>0$ and $ω(x0)≠0$ for some $x_0$.