Lecture Notes on Euler's Theorem for Homogeneous Functions

Introduction

• Topic: Euler's theorem for homogeneous functions
• Focus on defining homogeneous functions and proving Euler's theorem.

Homogeneous Functions

Definition

• A function f(x, y) is homogeneous of degree n if:
  [ f(\alpha x, \alpha y) = \alpha^n f(x, y) ]
  When applying this to a function of two variables.

Examples

• Example 1:

  • Function: [ f(x, y) = x^2 + y^2 ]
  • Homogeneous of degree 2: [ f(\alpha x, \alpha y) = \alpha^2 f(x, y) ]

• Example 2:

  • Function: [ f(x, y) = \sin^{-1}(\frac{x}{y}) ]
  • Homogeneous of degree 0: [ f(\alpha x, \alpha y) = f(x, y) ]

• Example 3:

  • Function: [ f(x, y) = \frac{x^4 + y^4}{\sqrt{x} + \sqrt{y}} ]
  • Homogeneous of degree ( \frac{7}{2} )

• Non-Homogeneous Example:

  • Function: [ f(x, y) = x^3 + xy ]
  • Not homogeneous as degrees do not match after substitution.

Euler's Theorem

Theorem Statement

• If f is homogeneous of degree n and has continuous first and second order partial derivatives, then:
  1. [ x f_x + y f_y = n f ]
  2. Further differentiation leads to a second relationship.

Proof

1. Assume f is homogeneous of degree n, i.e., [ f(x, y) = x^n g(\frac{y}{x}) ].
2. Find partial derivatives:
   • [ f_x = n x^{n-1} g(\frac{y}{x}) + x^n g'(\frac{y}{x}) ]
   • [ f_y = x^n g' ]
3. Substitute into Euler's equation: [ x f_x + y f_y = n f ]

Second Result

• Differentiate both sides with respect to x and y to obtain: [ x^2 f_{xx} + y^2 f_{yy} + 2xy f_{xy} = n(n-1)f ]_

Application of Euler's Theorem

Example Problems

1. First Problem:

   • Function: [ u = \sqrt{y^2 - x^2} \sin^{-1}(\frac{x}{y}) ]
   • Homogeneous of degrees 1 and 0.
   • Apply Euler's theorem: [ x u_x + y u_y = n u ]

2. Second Problem:

   • Function: [ u = \frac{y^3 - x^3}{y^2 + x^2} ]
   • Homogeneous of degree 1.
   • Euler's theorem gives the result immediately.

3. Non-Homogeneous Example:

   • Function: [ u = \ln( x^4 + y^4 ) ]
   • Not homogeneous, but can manipulate to apply properties of homogeneous functions to find derivatives.

Consequence of Euler's Theorem

• If f is a sum of two homogeneous functions of different degrees, the theorem can still apply:
  [ x f_x + y f_y = mf + nh ]
  where m and n are the degrees of the respective functions.

Conclusion

• Euler's theorem is a powerful tool for analyzing homogeneous functions and their derivatives.
• Applications in various examples demonstrate the theorem's utility.

Final Notes

• Encourage practice with a variety of functions to solidify understanding of the theorem and its applications.