[Math] If $\omega$ is an $n$-form on a compact $n$-manifold $M$ without boundary, then $\omega $ is exact if and only if $\int\limits_{M}\omega=0$

Question

If $\omega$ is an $n$-form on a compact $n$-dimensional manifold $M$ without boundary, then $\omega $ is exact if and only if $\int\limits_{M}{\omega }=0$.

Maybe there are two ways – use de Rham theory, and another way is to prove this directly.I don't know both. Help!

Answer 1

(Is $M$ orientable?)

If $\omega$ ist exact, then $$ \int\limits_{M}{\omega }= \int\limits_{M}{d\theta }=\int\limits_{\partial M}{\theta } = \int\limits_{\emptyset}\theta=0 $$ If $\int\limits_{M}{\omega }=0$ (or for the above direction as well), you can apply the fact that $\int\limits_{M}:H^n\rightarrow \mathbb R$ is an isomorphism.