Change of integration variables in multi-dimensional integral I have the following two-dimensional integral at hand
\begin{equation}
\int_{0}^{\infty}dx_{1}\int_{0}^{x_{1}}dx_{2} F(x_{1}-x_{2},x_{2}),
\end{equation}
where $F(x,y): \mathbb{R}^{2} \rightarrow \mathbb{R}$ is an arbitrary scalar function on a two-dimensional space.
Intuitively, by imagining the integration domain it is clear that this should be equal to
\begin{equation}
\int_{0}^{\infty}dx_{1}'\int_{0}^{\infty}dx_{2}' F(x_{1}',x_{2}').
\end{equation}
How can I show this formally via a change of integration variables?

Since $F$ is integrated on$$\{(x_1,\,x_2)\in\Bbb R|0\le x_2\le x_1\}=\{(x_1,\,x_2)\in\Bbb R|x_1-x_2\ge0\land x_2\ge0\},$$the change of variables $x_1^\prime:=x_1-x_2,\,x_2^\prime:=x_2$ rewrites the integral as$$\int_0^\infty dx_1^\prime\int_0^\infty dx_2^\prime JF(x_1^\prime,\,x_2^\prime),$$with $J$ a Jacobian determinant we need to check is $1$. I'll leave that part to you.
