https://file.aiquickdraw.com/mathbot/file/1726322564269wqwg1vsr.jpg

The equation provided in the image is:

\[
e = x^2 + y + z^2
\]

We are asked to prove that:

\[
x e_{x} y_{z} = y e_{y} z_{x} = 2xyz
\]

To proceed with the proof, we need to first determine the partial derivatives of \( e \) with respect to \( x \), \( y \), and \( z \), and then analyze the expressions accordingly.

### Step 1: Partial derivatives
Given \( e = x^2 + y + z^2 \), let's compute the partial derivatives:

- \( e_{x} = \frac{\partial e}{\partial x} = 2x \)
- \( e_{y} = \frac{\partial e}{\partial y} = 1 \)
- \( e_{z} = \frac{\partial e}{\partial z} = 2z \)

### Step 2: Interpretation of notation
From the problem statement, \( e_{x} \), \( e_{y} \), and \( e_{z} \) refer to the partial derivatives of \( e \) with respect to \( x \), \( y \), and \( z \), respectively. The subscripted variables such as \( y_z \) and \( z_x \) seem to imply functional dependencies or mixed partials, but based on the standard interpretation of the equation, it could be symbolic of certain relationships to test between these terms.

However, I would need to interpret further to clarify. Can you clarify if there is a specific notation rule or further context on these terms in your problem?

Let me know if you want more detailed steps in expanding this proof.

In the meantime, consider the following questions:

1. How are \( y_z \) and \( z_x \) defined or derived from the initial condition?
2. Is there a specific geometric or physical interpretation for the variables?
3. Could there be any missing functional relationships between the variables?
4. Are there any symmetries in the problem that can simplify the proof?
5. Could the result involve higher-order derivatives?

### Tip:
When working with partial derivatives in multivariable functions, always ensure you have all dependencies between variables clearly defined, as it can significantly alter the final result.

If e = x^2 + y + z^2, prove that x e_x y_z = y e_y z_x = 2xyz.

Learn how to prove the relation x e_x y_z = y e_y z_x = 2xyz using partial derivatives from the equation e = x^2 + y + z^2. Detailed step-by-step solution provided for university-level multivariable calculus.

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