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

Solve the two algebraic constraints for the squared coordinates:On any nonsingular real branch, implicit differentiation at fixed $u$ gives $(\partial x/\partial z{)}_u=z/(3x)$ and $(\partial y/\partial z{)}_u=z/(3y)$. Differentiating $z^2=2x^2+y^2$ with the other original coordinate fixed gives $(\partial z/\partial x{)}_y=2x/z$ and $(\partial z/\partial y{)}_x=y/z$. ThereforeThis is the multivariable chain rule applied to $z(x(z,u),y(z,u))=z$. These formulas assume the relevant denominators are nonzero and the chosen variables are valid local coordinates.