Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 19 / c / Solution

For an independent inversion, use the coordinate relationsThe positive-octant domain fixes their angle branches unambiguously. Let $s=\sqrt{x^2+y^2}=W\tan \theta$. Differentiating at fixed Cartesian coordinates givesand ${\varphi }_x=-y/s^2=-\sin \varphi /(W\tan \theta )$, ${\varphi }_y=x/s^2=\cos \varphi /(W\tan \theta )$, ${\varphi }_z=0$. Hence the inverse chain rule givesandMultiplying these coefficients by the forward coefficient matrix gives the identity. The original PDF correctly holds $x,y$ fixed in the final derivative; the transcription's $x,z$ subscript there is a typo.