lets assume i haven given two functions $f:\mathbb{R}^m\rightarrow \mathbb{R}$ and $h:\mathbb{R}^n\rightarrow \mathbb{R}$ (m < n) and a surjective mapping $g:\mathbb{R}^n \rightarrow \mathbb{R}^m$
I would like to compute the following integral: $$ \int f(g(y))h(y) \;dy $$
for this, i would like to perform a variable substitution $x=g(y)$. However, since g is surjective, there is a set $\Omega(x)=\{y| g(y)=x\}$. Unfortunately, I never learned how to treat this case and people turn away when i ask how to do this :)
My naive approach was: $$ \int f(g(y))h(y) \;dy = \int f(x) \left[\int_{\Omega(x)}h(y)\;dy\right] \; dx $$
However, when i test this with functions where i know what i should obtain, (e.g. $y \in \mathbb{R}^2$, $f(g(y))=\exp(-\lVert y\rVert^2/2)$ with $g(y)=\lVert y\rVert^2$ and $h(y)=1$) I observe that i seem to be missing a term.
I get as circumference of the circle of radius $\sqrt x$:
$$\left[\int_{\Omega(x)}h(y)\;dy\right]=2\pi \sqrt{x}$$ and substituting this, i get
$$ \int_0^\infty f(x) 2\pi \sqrt{x} \; dx = 2\pi \sqrt{2\pi} $$
so I seem to be off by $2\pi$