A (not so?) simple question about differential forms

Question

Let $M^n$ be a compact orientable manifold and let $\omega$ be a $(n-1)$-form in $M^n$. I want to show that there is $p\in M$ such that $(d\omega)_p=0$.

Can somebody help me, please ? Thanks :)

Answer 1

Hint: Use the Stokes formula for integrals.

Well, suppose not. Then $\mathrm{d} \omega$ is a nowhere-vanishing differential $n$-form, so one of $\alpha = \mathrm{d} \omega$ or $\alpha = -\mathrm{d} \omega$ is a volume form. (I assume $M$ is connected.) But then $\int_{M} \alpha > 0$, contradicting the Stokes formula $\int_{M} \mathrm{d} \omega = \int_{\partial M} \omega = 0$ (because $\partial M = \emptyset$).