= Ordered-cone substitution by products and ratios
{title2=$(\alpha,\beta,\gamma)=(x/y,xy,yz)$}
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 $x\,dx\,dy\,dz=\tfrac12\,d\alpha\,d\beta\,d\gamma$. 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>.
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