Verify Euler's theorem for the function \( u=x^{2}+y^{2}+2 x y \)

Euler's theorem states that for a homogeneous function \(u(x, y)\) of degree \(n\), the following relationship holds:

\[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u \]

First, let's verify if \( u = x^2 + y^2 + 2xy \) is a homogeneous function and determine its degree.

A function \(u(x, y)\) is homogeneous of degree \(n\) if:

\[ u(\lambda x, \lambda y) = \lambda^n u(x, y) \]

Let's test \(u\):

\[ u(\lambda x, \lambda y) = (\lambda x)^2 + (\lambda y)^2 + 2 (\lambda x)(\lambda y) \]

\[ = \lambda^2 x^2 + \lambda^2 y^2 + 2 \lambda^2 xy \]

\[ = \lambda^2 (x^2 + y^2 + 2xy) \]

This shows that \( u(\lambda x, \lambda y) = \lambda^2 u(x, y) \), so \(u(x, y) = x^2 + y^2 + 2xy\) is a homogeneous function of degree 2.

Next, we compute the partial derivatives of \(u\):

\[ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2 + 2xy) = 2x + 2y \]

\[ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^2 + 2xy) = 2y + 2x \]

Now, we check Euler's theorem:

\[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \]

Substitute the partial derivatives:

\[ x (2x + 2y) + y (2y + 2x) \]

\[ = 2x^2 + 2xy + 2y^2 + 2xy \]

\[ = 2x^2 + 2y^2 + 4xy \]

Notice that:

\[ u = x^2 + y^2 + 2xy \]

So,

\[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2(x^2 + y^2 + 2xy) = 2u \]

Therefore, Euler's theorem is verified for the function \( u = x^2 + y^2 + 2xy \) with \( n = 2 \).

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:
1. What is a homogeneous function in general terms?
2. How can Euler's theorem be applied to functions of more than two variables?
3. Can Euler's theorem be used for non-homogeneous functions?
4. What is the geometric interpretation of homogeneous functions?
5. How does the degree of a homogeneous function affect its properties?
6. Can you provide an example of a homogeneous function of degree 3?
7. How would you verify Euler's theorem for the function \( u = x^3 + y^3 \)?
8. How does Euler's theorem relate to the concept of scaling in physics and engineering?

**Tip:** When verifying Euler's theorem, always start by confirming the function's degree of homogeneity before applying the partial derivative operations.

Explore how Euler's theorem is verified for the function u = x^2 + y^2 + 2xy through detailed mathematical analysis. Learn about homogeneous functions, partial derivatives, and the application of Euler's theorem. Suitable for advanced undergraduate students and mathematicians.

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