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The equation in the image relates to a transformation of variables in the context of partial derivatives, a common technique in multivariable calculus and mathematical physics. Let's examine and solve the equation step-by-step.

The given relations are:
- \( x = 3u - v \)
- \( y = v - 3u \)
- \( \frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = m \frac{\partial z}{\partial x} + n \frac{\partial z}{\partial y} \)

The goal is to find values for \( m \) and \( n \) such that the equation holds. This involves using the chain rule for partial derivatives.

### Applying Chain Rule:
Given \( z = f(x, y) \), then
\[ \frac{\partial z}{\partial u} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial u} \]
\[ \frac{\partial z}{\partial v} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial v} \]

Using the given transformations:
- \( \frac{\partial x}{\partial u} = 3 \)
- \( \frac{\partial x}{\partial v} = -1 \)
- \( \frac{\partial y}{\partial u} = -3 \)
- \( \frac{\partial y}{\partial v} = 1 \)

Substituting these into the derivatives of \( z \):
\[ \frac{\partial z}{\partial u} = 3 \frac{\partial f}{\partial x} - 3 \frac{\partial f}{\partial y} \]
\[ \frac{\partial z}{\partial v} = -\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \]

Adding these:
\[ \frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = (3 \frac{\partial f}{\partial x} - 3 \frac{\partial f}{\partial y}) + (-\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}) = 2 \frac{\partial f}{\partial x} - 2 \frac{\partial f}{\partial y} \]

### The Given Equation:
\[ 2 \frac{\partial f}{\partial x} - 2 \frac{\partial f}{\partial y} = m \frac{\partial f}{\partial x} + n \frac{\partial f}{\partial y} \]

To solve for \( m \) and \( n \), equate the coefficients:
- From \( \frac{\partial f}{\partial x} \) terms: \( 2 = m \)
- From \( \frac{\partial f}{\partial y} \) terms: \( -2 = n \)

### Conclusion:
The values are \( m = 2 \) and \( n = -2 \). 

If you have any questions or need further explanations, feel free to ask! Here are some follow-up queries that might interest you:

1. How can we generalize the method used here to other transformations?
2. What is the geometric interpretation of these transformations in the context of vector fields?
3. Could you explain how these calculations change if the transformations are non-linear?
4. How does this relate to Jacobian matrices in coordinate transformations?
5. What other applications might such transformations and derivative relations have in engineering or physics?

**Tip:** When dealing with transformations, always consider the chain rule for derivatives to relate changes in different coordinate systems or dimensions.

The equation in the image relates to a transformation of variables in the context of partial derivatives, a common technique in multivariable calculus and mathematical physics.

Learn how to solve a transformation involving partial derivatives and the chain rule with detailed steps. This problem involves transformations of variables x = 3u - v and y = v - 3u and the calculation of the corresponding m and n values.

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