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

Apply the multivariable chain rule to an arbitrary differentiable scalar function of $x,y,z$. With the subscripts indicating the coordinates held fixed, the first operator isHolding $W,\varphi$ fixed when differentiating with respect to $\theta$ givesFinally, holding $W,\theta$ fixed givesFor example, the coefficient of ${\partial }_x$ in ${\partial }_{\theta }$ is precisely $(\partial x/\partial \theta {)}_{W,\varphi }$; all coefficients follow by differentiating the given coordinate functions.