\( w=f(u, v), u=x+y, v=x-y \), sonow that \( \frac{\partial w}{\partial x} \frac{\partial w}{\partial y}=\left(\frac{\partial w}{x y}\right)^{2}-\left(\frac{0 w}{0 v}\right)^{2} \)

To solve the problem, we need to find the partial derivatives of \( w \) with respect to \( x \) and \( y \) using the chain rule, given the relationships \( u = x + y \) and \( v = x - y \). 1. **Find \( \frac{\partial w}{\partial x} \)**: Using the chain rule: \[ \frac{\partial w}{\partial x} = \frac{\partial w}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial w}{\partial v} \frac{\partial v}{\partial x} \] Calculating the partial derivatives of \( u \) and \( v \): \[ \frac{\partial u}{\partial x} = 1, \quad \frac{\partial v}{\partial x} = 1 \] Thus, \[ \frac{\partial w}{\partial x} = \frac{\partial w}{\partial u} \cdot 1 + \frac{\partial w}{\partial v} \cdot 1 = \frac{\partial w}{\partial u} + \frac{\partial w}{\partial v} \] 2. **Find \( \frac{\partial w}{\partial y} \)**: Similarly, using the chain rule: \[ \frac{\partial w}{\partial y} = \frac{\partial w}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial w}{\partial v} \frac{\partial v}{\partial y} \] Calculating the partial derivatives of \( u \) and \( v \): \[ \frac{\partial u}{\partial y} = 1, \quad \frac{\partial v}{\partial y} = -1 \] Thus, \[ \frac{\partial w}{\partial y} = \frac{\partial w}{\partial u} \cdot 1 + \frac{\partial w}{\partial v} \cdot (-1) = \frac{\partial w}{\partial u} - \frac{\partial w}{\partial v} \] 3. **Now compute \( \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \)**: \[ \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} = \left( \frac{\partial w}{\partial u} + \frac{\partial w}{\partial v} \right) \left( \frac{\partial w}{\partial u} - \frac{\partial w}{\partial v} \right) \] Using the difference of squares: \[ = \left( \frac{\partial w}{\partial u} \right)^2 - \left( \frac{\partial w}{\partial v} \right)^2 \] 4. **Now, let's analyze the right-hand side of the equation**: The right-hand side of the equation is given as: \[ \left( \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \right) = \left( \frac{\partial w}{\partial u} \right)^2 - \left( \frac{\partial w}{\partial v} \right)^2 \] 5. **Final equation**: The equation you provided seems to have some notation that is not standard, particularly \( \frac{0 w}{0 v} \). Assuming it means \( \frac{\partial w}{\partial v} \), we can conclude that: \[ \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} = \left( \frac{\partial w}{\partial u} \right)^2 - \left( \frac{\partial w}{\partial v} \right)^2 \] This confirms that the left-hand side equals the right-hand side, as derived. Thus, the equation holds true under the assumptions made.