Let $\alpha$, $\beta$ be two $r$-forms continuous in $U\subseteq \mathbb{R}^n$ open.
If $\int_M \alpha =\int_M \beta $ for all surface $M\subseteq U$ dimension $r$, compact, with boundary, then show that $\alpha=\beta$.
But if $\alpha-\beta\neq 0$ then $\int_M\alpha-\beta \neq 0$ always? No matter who it is $M$? I.e., is the condition "for all" $M$ necessary?