Let $\omega$ a $C^1$, $r$-form in $\mathbb{R}^n$ such that $\int_M\omega=0$ for all orientable compact manifold $M$ of dimension $r$ in $\mathbb{R}^n$, prove that $\omega$ is closed.
I need to use the Stokes Theorem in this problem. What I had originally thought is not correct, as discussed in the comments, I will leave it for future reference.
What I thought now was to use pullback, but I have some doubts if it all makes sense.
Let $f^{*}:\mathbb{R}^n\to\mathbb{R}^{r+1}$ the pullback (this make sense?). So, $f^{*}\omega$ is a $r$-form in $\mathbb{R}^{r+1}$ and $d(f^{*}\omega)=f^{*}(d\omega)$ is a $(r+1)$-form in $\mathbb{R}^{r+1}$.
So, if I guarantee that $f$ is a diffeomorphism, $f^*$ is an isomorphism of vector-spaces of $(r+1)$-forms, hence $0=df^*\omega = f^*d\omega$ implies $d\omega=0$.
So, let's assume that $f^{*}(d\omega)\neq 0$, as $f^{*}(d\omega)$ is a $(r+1)$-form in $\mathbb{R}^{r+1}$ it makes sense to speak of positivity of the form, so let's call $\Omega$ an open ball where $f^{*}(d\omega)(x)>0$. So, $\int_{\Omega} f^{*}(d\omega)>0$, but
$$\int_{\Omega} f^{*}(d\omega)= \int_{\partial \Omega} f^{*}\omega \stackrel{?}{=}0$$
So, for this proof to be right, I need to define this pullback well so that it is an isomorphism, can anyone help me with this? And yet, having to ensure that the above equality is zero, is that true?
Let $B$ be the $r+1$-dimensional ball in $\mathbb{R}^n$, we know that the boundary of $B$ is the $r$-dimensional sphere, which is compact and orientable. So by the stokes theorem we have, $$\int_B d\omega=\int_{\partial B} \omega=0\Longrightarrow \int_B d\omega=0$$ Now, to conclude that $\omega$ is closed I thought to assume, that there is a region where $\omega$ is positive (or negative). Let $\Omega=\{x\in B:\omega(x)>0\}$. So, I believe this is true - $\partial\Omega=\emptyset$ or - $\partial\Omega$ is compact and orientable In both cases, applying Stokes again, we have $$\int_\Omega d\omega=\int_{\partial \Omega} \omega=0$$ That would be absurd, because we would have to have $\int_\Omega \omega >0$. My doubts about my solution are whether my set $B$ really satisfies the hypothesis of the problem, I know it satisfies if $B$ is in $\mathbb{R}^{r+1}$, but being in $\mathbb{R}^n$ also satisfies? And my biggest doubt is about the statements I made about the $\Omega$ boundary, intuitively seems correct, but I don't know any results that say that.