https://file.aiquickdraw.com/mathbot/file/17273568062934zv21ae8.jpg

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?

Prove the relationship involving second-order partial derivatives for the function z=f(x,y), where x=u^2−v^2 and y=2uv, and show that (∂^2z/∂u^2 + ∂^2z/∂v^2) = 4(∂^2z/∂x^2 + ∂^2z/∂y^2).

Explore how to prove the second-order partial derivative relationship for z=f(x,y) where x=u^2−v^2 and y=2uv. This problem involves multivariable calculus concepts, including the chain rule, second-order derivatives, and their transformations.

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