The problem provided involves a function $z = f(x, y)$ with variables $x$ and $y$ defined in terms of new variables $u$ and $v$. The goal appears to be proving some relationship involving partial derivatives.

From the image:

1. $z = f(x, y)$, where $x = u^2 - v^2$ and $y = 2uv$.
2. The task seems to involve proving a relationship related to second-order partial derivatives, possibly involving the Laplacian operator or a similar equation.

The expression to be proved involves second-order partial derivatives of $z$ with respect to $u$ and $v$: $\frac{\partial^2 z}{\partial u^2} + \frac{\partial^2 z}{\partial v^2} = 4 \left( \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} \right).$

This suggests you are tasked with finding and verifying this result by computing the partial derivatives of $z$ with respect to $u$ and $v$, using the chain rule to express them in terms of the second-order partial derivatives with respect to $x$ and $y$.

Approach

1. Use the chain rule for multivariable functions to express the partial derivatives of $z$ with respect to $u$ and $v$ in terms of $x$ and $y$.
2. Compute the second-order partial derivatives with respect to $u$ and $v$.
3. Compare these with the second-order partial derivatives with respect to $x$ and $y$.
4. Simplify and prove the equation given in the problem.

Would you like me to proceed with the detailed step-by-step solution?