Let $\omega_1,\omega_2$ be two n-forms on a $n$-dimensional manifold $M$.
Now, imagine we have for every open $N \subset M$ that
$$\int_{N}\omega_1 = \int_N \omega_2.$$
Can anybody show me how to prove that both forms are equal?- I suspect that this is true as it sounds natural, although I don't know for sure.
Define a $n$-form $\omega :=\omega_1-\omega_2$ so that $$ \omega = f\nu $$ where $\nu$ is a volume form on $M$.
If $f(p)>0$ so $f|_{B_\epsilon (p)} > 0$ Then $$ 0=\int_{B_\epsilon (p)} \omega $$ Hence we conclude that $f=0$.