Let $z$ be a function of $u$ and $v$ where $u=x+y$ and $v=3x-3y$.
I have previously shown, by the Chain Rule, that $$\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}= \left(\frac{\partial z}{\partial u}\right)^2 - 9 \left(\frac{\partial z}{\partial v}\right)^2.$$
Now, "assuming equality of mixed second-order partial derivatives", I must show that $$\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = 2\frac{\partial^2 z}{\partial u^2} + 18\frac{\partial^2 z}{\partial v^2}.$$
Firstly, I don't understand what is meant by "equality of mixed second-order partial derivatives". Please could somebody explain this?
I think I must use the Chain Rule again to get this equation but I don't really understand how to do this. I just need a pointer to get me started. Thanks.