Past exam of the mathematics course of the University of Cambridge / 2005 / ia / Paper 3 / 12A (Vector Calculus)

The substitution is nonsingular in the interior $x>0$, $0<y<1$, $0<z<x$. Its inverse isThus $y<1$ is equivalent to $\beta <\alpha$, while $z<x$ is equivalent to $\gamma <xy=\beta$. Positivity gives the full transformed domainThe forward Jacobian determinant isTherefore $xdxdydz=\frac{1}{2}d\alpha d\beta d\gamma$. The boundary faces where the formula fails have measure zero. The change of variables formula givesThe integrand is nonnegative, so Tonelli's theorem permits reordering the integration even before convergence has been established. Integrating first in $\alpha$, then in $\beta$, givesThe assumptions $A,B,C>0$ make all three integrals of exponential functions converge. Equivalently, the ordered-cone substitution by products and ratios can be followed by the increment variables $\alpha -\beta$, $\beta -\gamma$, $\gamma$, all independently positive and with Jacobian determinant one; the exponent then separates into three linear terms with coefficients $A$, $A+B$ and $A+B+C$.