I have an integral of a function that depends on multiple variables:
$\int_\Omega f(x,y,z)dzdydx=1$
where $x, y$, and $z$ can take values from $-\infty$ to $\infty$. Now each of the variables are functions of other three variables $u, v,$ and $w$.
$x = x(u,v,w), y=y(u,v,w), z=z(u,v,w)$
I know that the variables $u, v $ and $w$ can take values from $0$ to $1$, from $0$ to $\infty$ and from $0$ to $\infty$ respectively. However, these functions are not injective and the inverse transformation
$(u,v,w)=(u(x,y,z),v(x,y,z),w(x,y,z))$
is not analytical. My question is, what is the integrand that I have to use in order to do this integral using the variables $(u,v,w)$? In other words, in the following equation,
$\int_\Omega f(x,y,z)dzdydx=\int_\Gamma g(u,v,w)dudvdw$
How can I find the function $g(u,v,w)$?