Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 9 / b / Solution

Differentiate both algebraic constraints at fixed $y$ or fixed $x$:The multivariable chain rule for $\varphi (x,y)=\psi (z,u)$ therefore givesDivide the first identity by $x$ and the second by $y$, and add. The ${\psi }_u$ terms cancel, leavingThe Jacobian determinant of $(x,y)\mapsto (z,u)$ is $-3xy/(zu)$, so the calculation is valid on real branches with $xyzu\ne 0$. This explicitly supplies the local independence assumption underlying the change of coordinates.