https://r2.erweima.ai/mathbot/image/1727049866146omd6q2xs.jpg

The function given in the image is:

\[
u(x, y) = e^x (x \cos y - y \sin y)
\]

You are asking about **harmonic functions**, so the question is likely whether this function \( u(x, y) \) is harmonic. A function is **harmonic** if it satisfies **Laplace's equation**:

\[
\Delta u = 0
\]

where \( \Delta u \) is the **Laplacian** of \( u \), given by:

\[
\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}
\]

### Steps:
1. Find \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \)
2. Differentiate again to find \( \frac{\partial^2 u}{\partial x^2} \) and \( \frac{\partial^2 u}{\partial y^2} \)
3. Add them to check if \( \Delta u = 0 \).

I'll proceed by computing the partial derivatives and the Laplacian.

Let's calculate this.The Laplacian of the function \( u(x, y) = e^x (x \cos y - y \sin y) \) is:

\[
\Delta u = 0
\]

This confirms that the function is **harmonic**, as it satisfies Laplace's equation.

Would you like further details on the derivation or explanation of harmonic functions? Let me know!

Here are 5 related questions you might find helpful:
1. What are some other examples of harmonic functions in two variables?
2. How does Laplace's equation apply to physical systems (e.g., in fluid dynamics or electrostatics)?
3. What is the relationship between harmonic functions and potential theory?
4. Can a function be harmonic in one region but not in another?
5. What are the boundary conditions typically applied to problems involving harmonic functions?

**Tip:** Harmonic functions have the property of averaging, meaning that the value of the function at any point is the average of its values in a surrounding region. This makes them useful in physics and engineering, especially in steady-state conditions.

Is the function u(x, y) = e^x (x cos y - y sin y) harmonic?

Explore whether the function u(x, y) = e^x (x cos y - y sin y) is harmonic by checking if it satisfies Laplace's equation. This page explains key concepts like harmonic functions, Laplace's equation, and how to compute partial derivatives.

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