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

Suppress the held-fixed subscripts only within this matrix display. Collecting the preceding operator coefficients givesIf $J=\partial (x,y,z)/\partial (W,\theta ,\varphi )$ uses the usual convention of coordinate functions as rows, the displayed coefficient matrix is $J^{\mathsf{T}}$. Its determinant is $W^2{\sec }^2\theta \tan \theta >0$ on the given domain, so the inverse operator transformation exists everywhere there.