= 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 yields
$$
\boxed{\left(\frac{\partial w}{\partial x}\right)_y
\left(\frac{\partial w}{\partial y}\right)_x
=\left(\frac{\partial w}{\partial u}\right)_v^2-
\left(\frac{\partial w}{\partial v}\right)_u^2.}
$$
The subscripts specify the independent coordinate held fixed; they are necessary because changing coordinates changes which direction a partial derivative follows.
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