Ordered-cone substitution by products and ratios ($(\alpha ,\beta ,\gamma )=(x/y,xy,yz)$)

On $x>0$, $0<y<1$, $0<z<x$, this change of variables is a bijection onto $0<\gamma <\beta <\alpha$. Its inverse is $x=\sqrt{\alpha \beta }$, $y=\sqrt{\beta /\alpha }$, $z=\gamma \sqrt{\alpha /\beta }$, and its forward Jacobian determinant is $2x$. Thus $xdxdydz=\frac{1}{2}d\alpha d\beta d\gamma$. Passing to increments $(\alpha -\beta ,\beta -\gamma ,\gamma )$ then maps the ordered cone to the positive octant, turning an exponential with linear exponent into a product of three one-dimensional integrals.