Let $M$ be a compact $m$ dimensional manifold with boundary $\partial M$.

Assume that $I_{1}, I_{2}$ are two linear functionals on $\Omega^{m}(M), \Omega^{m-1}(\partial M)$, respectively. Assume that we have $I_{1}(d\alpha )=I_{2} ( \alpha )$ for every $m-1$ differential form on $M$.

Are $I_{1},I_{2}$ necessarily equal to a constant multiple of the usual integrals?

Let $M$ be a compact $m$ dimensional manifold with boundary $\partial M$.

Assume that $I_{1}, I_{2}$ are two linear functionals on $\Omega^{m}(M), \Omega^{m-1}(\partial M)$, respectively. Assume that we have $I_{1}(d\alpha )=I_{2} ( \alpha )$ for every $m-1$ differential form $\alpha$ on $M$.

Are $I_{1},I_{2}$ necessarily equal to a constant multiple of the usual integral?