\( \omega=f(c, v), u=x+y, v=x-y \), show that \( \frac{\partial u}{\partial x} \frac{2 w}{\partial y}=\left(\frac{3 w}{b y}\right)^{2}-\left(\frac{0 w}{0 u}\right)^{2} \)

To show the equation \[ \frac{\partial u}{\partial x} \frac{2 w}{\partial y} = \left(\frac{3 w}{b y}\right)^{2} - \left(\frac{0 w}{0 u}\right)^{2}, \] we first need to analyze the components of the equation. 1. **Understanding the Variables**: - We have \( u = x + y \) and \( v = x - y \). - The function \( \omega = f(c, v) \) suggests that \( \omega \) depends on some variables \( c \) and \( v \). 2. **Calculating Partial Derivatives**: - The partial derivative of \( u \) with respect to \( x \) is: \[ \frac{\partial u}{\partial x} = \frac{\partial (x + y)}{\partial x} = 1. \] 3. **Finding \( \frac{\partial w}{\partial y} \)**: - To find \( \frac{\partial w}{\partial y} \), we need to express \( w \) in terms of \( y \). Since \( w \) is a function of \( c \) and \( v \), we need to know how \( c \) relates to \( y \) and \( v \). However, without additional information about \( c \), we cannot compute \( \frac{\partial w}{\partial y} \) directly. 4. **Analyzing the Right Side of the Equation**: - The term \( \left(\frac{0 w}{0 u}\right)^{2} \) is problematic because it suggests a division by zero, which is undefined. This indicates that the right-hand side of the equation may not be well-defined unless \( w \) or \( u \) is zero. 5. **Simplifying the Equation**: - The left-hand side simplifies to: \[ \frac{\partial u}{\partial x} \frac{2 w}{\partial y} = 1 \cdot \frac{2 w}{\partial y} = \frac{2 w}{\partial y}. \] - The right-hand side simplifies to: \[ \left(\frac{3 w}{b y}\right)^{2} - \left(\frac{0 w}{0 u}\right)^{2} = \left(\frac{3 w}{b y}\right)^{2} - \text{undefined}. \] 6. **Conclusion**: - The equation as presented cannot be shown to hold true without additional context or definitions for \( w \), \( c \), and how they relate to \( y \) and \( u \). The presence of \( \left(\frac{0 w}{0 u}\right)^{2} \) suggests that there may be a mistake or a need for clarification in the formulation of the problem. In summary, without further information or context, we cannot demonstrate the validity of the equation as it stands.