Math Problem Statement

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

Solution

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

xux+yuy=nux \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u

First, let's verify if u=x2+y2+2xyu = x^2 + y^2 + 2xy is a homogeneous function and determine its degree.

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

u(λx,λy)=λnu(x,y)u(\lambda x, \lambda y) = \lambda^n u(x, y)

Let's test uu:

u(λx,λy)=(λx)2+(λy)2+2(λx)(λy)u(\lambda x, \lambda y) = (\lambda x)^2 + (\lambda y)^2 + 2 (\lambda x)(\lambda y)

=λ2x2+λ2y2+2λ2xy= \lambda^2 x^2 + \lambda^2 y^2 + 2 \lambda^2 xy

=λ2(x2+y2+2xy)= \lambda^2 (x^2 + y^2 + 2xy)

This shows that u(λx,λy)=λ2u(x,y)u(\lambda x, \lambda y) = \lambda^2 u(x, y), so u(x,y)=x2+y2+2xyu(x, y) = x^2 + y^2 + 2xy is a homogeneous function of degree 2.

Next, we compute the partial derivatives of uu:

ux=x(x2+y2+2xy)=2x+2y\frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2 + 2xy) = 2x + 2y

uy=y(x2+y2+2xy)=2y+2x\frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^2 + 2xy) = 2y + 2x

Now, we check Euler's theorem:

xux+yuyx \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}

Substitute the partial derivatives:

x(2x+2y)+y(2y+2x)x (2x + 2y) + y (2y + 2x)

=2x2+2xy+2y2+2xy= 2x^2 + 2xy + 2y^2 + 2xy

=2x2+2y2+4xy= 2x^2 + 2y^2 + 4xy

Notice that:

u=x2+y2+2xyu = x^2 + y^2 + 2xy

So,

xux+yuy=2(x2+y2+2xy)=2ux \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=x2+y2+2xyu = x^2 + y^2 + 2xy with n=2n = 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=x3+y3u = 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.

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Math Problem Analysis

Mathematical Concepts

Homogeneous functions
Partial derivatives
Euler's theorem

Formulas

Euler's theorem: x * du/dx + y * du/dy = n * u

Theorems

Euler's theorem for homogeneous functions

Suitable Grade Level

Advanced undergraduate level