However, the last part confuzes me....

If $\displaystyle f(x,y)=f(u,v)$, show that:

$\displaystyle \frac{\partial^{2} f}{\partial x \partial y}=ab (\frac{\partial^{2} f}{\partial u^{2}} - \frac{\partial^{2} f}{\partial v^{2}}) + (b^{2}-a^{2})\frac{\partial^{2} f}{\partial u \partial v} $

The problem is i'm not sure what $\displaystyle f(x,y)=f(u,v)$ actually means. I understand it to be: f is a function of two variables, x and y, and f is a function of two variables, u and v, and these two functions are equal. But u and v are themselves functions of x and y so what is f?