The function given is:

$u = \arctan\left(\frac{x^2 + y^2}{x + y}\right)$

If you need to explore or differentiate this function, let's outline a few possible steps depending on what you're looking for:

1. Partial derivatives:

• To find the partial derivative of $u$ with respect to $x$ or $y$, we use the chain rule. Let’s compute these if that’s what you need.

2. Gradient of $u$:

• The gradient of $u$ will be a vector formed by $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$.

3. Level Curves:

• Level curves $u = \text{constant}$ represent solutions to the equation $\frac{x^2 + y^2}{x + y} = \text{constant}$, which could describe certain curves in the plane.

Would you like to proceed with any of these steps, or is there something specific you'd like to explore about this function?

5 Relative Questions

1. Would you like to compute the partial derivatives of $u$ with respect to $x$ and $y$?
2. Are you interested in the gradient or directional derivatives of $u$?
3. Would you like to analyze the behavior of $u$ as $x$ and $y$ approach certain values?
4. Are you interested in visualizing the level curves of this function?
5. Would you like to apply this function to any specific values of $x$ and $y$?

Tip: When working with functions involving arctangents, recall that the range of $\arctan$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$, which can help analyze the behavior of the function.