Let $U$ an open set in $\Bbb{R}^m$, $f:U\rightarrow\Bbb{R}^m$ a $C^2$ map. Let $D\subset U$ a compact domain, with $C^2$ boundary $\partial D$. If $\left.f\right|_{\partial D}\equiv 0$, so $\int_{D}J(x)=0$, where $J(x)=\det f'(x)$.
$D$ is a compact domain, that is, $D$ is a compact surface with boundary. Since $\partial D$ is $C^2$, $D$ is $C^3$.
My problem here is that $J$ and $f$ are not differential forms. I want to apply the Stokes Theorem, but I dont know how.
What I tried: Let $\omega$ the volume form in $\Bbb{R}^m$. So $f^{*}\omega$ is a differential form in $U$. By hypothesis, $\left.f\right|_{\partial D}\equiv 0\implies \left.f^{*}\omega\right|_{\partial D}\equiv 0$. So, we have
$$\int_{\partial D}f^{*}\omega=0. $$
But, by Stokes theorem,
$$\int_{\partial D}f^{*}\omega=\int_{D}d(f^{*}\omega)=\int_{D}f^{*}(d\omega). $$
That is all that I got. I dont know if I am in the right way.