Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 2 / a / Solution

In a partial derivative, the variable held fixed matters. Holding $u$ fixed gives $x_v=-v$. To find $v_y$, hold $x$ fixed and invert the Jacobian matrix:ThusThe second formula holds on each local inverse branch away from $(u,v)=(0,0)$, where the Jacobian determinant vanishes. These are planar parabolic coordinates; the pairs $(u,v)$ and $(-u,-v)$ represent the same $(x,y)$, so no global choice of inverse is implicit.