We can perform integration by u-substitution like this:
$$ \int_D f(\mathbf{v})\mathrm{d}\mathbf{v}=\int_{D^\star} f(g(\mathbf{u}))\left|J_g\right|\mathrm{d}\mathbf{u} $$
Now if $h:\mathbb{R}^n\to \mathbb{R}$ I've always wondered if this is something like this is possible (written in $\mathbb{R}^2$):
$$ \iint_D f(x,y)\mathrm{d}x\mathrm{d}y=\int_\Gamma h(x,y)\mathrm{d}h(x,y) $$
Is this in some sense related to divergence theorem, because the second is an integral over a 1-form. Also, what is $\Gamma$ then? What's the general thing in $\mathbb{R}^n$?
Thanks.