Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 14 / a / ii / Solution

The coordinate partial derivatives are ${\xi }_y=1/\sqrt{y}$ and ${\eta }_y=-1/\sqrt{y}$. The multivariable chain rule therefore givesSince $\xi -\eta =4\sqrt{y}>0$, the expression entirely in the new coordinates isThe inverse coordinate formulas are $x=(\xi +\eta )/2$ and $y=(\xi -\eta {)}^2/16$, with domain $\xi >\eta$.