Past exam of the mathematical methods II course of the University of Oxford / 2017 / 7 / Solution

Holding $y$ fixed gives $u_x=v_x=1$, whereas holding $x$ fixed gives $u_y=1$ and $v_y=-1$. The chain rule therefore gives $w_x=w_u+w_v$ and $w_y=w_u-w_v$. Multiplication yieldsThe subscripts specify the independent coordinate held fixed; they are necessary because changing coordinates changes which direction a partial derivative follows.