Euler's Theorem

Euler's Theorem:

  • Euler's theorem for homogeneous functions provides a relationship between a homogeneous function and its partial derivatives.

Homogeneous Functions:

  • A function f(x,y) is homogeneous of degree n if it satisfies the following condition: f(tx,ty)=tnf(x,y) where t is a scalar and n is a constant.

Euler's Theorem Statement:

  • The theorem states that if f(x,y) is a homogeneous function of degree n, then: xfx+yfy=nf(x,y)

Proof of Euler's Theorem:

  • Start with the definition of a homogeneous function: f(tx,ty)=tnf(x,y)

  • Apply the chain rule to differentiate f(tx,ty) with respect to t: df(tx,ty)dt=ntn1f(x,y)

  • Chain rule for partial derivatives: fxdxdt+fydydt=ntn1f(x,y)

  • Substitute t=1: xfx+yfy=nf(x,y)

Applications of Euler's Theorem:

  • Economics and Utility Theory: Useful in analyzing production functions and utility functions in economics.
  • Physics and Engineering: Applied in problems involving laws of conservation and scaling phenomena.
  • Optimization and Gradient Descent: Helps in optimization problems involving homogeneous functions.

Special Cases:

  • Degree Zero Homogeneous Functions: For a function of degree zero, Euler's theorem doesn't apply as it won't satisfy the condition for homogeneity.

  • Multivariable Functions: The theorem extends to functions with more than two variables, following the same pattern of homogeneity.

Key Insights:

  • Relationship between Derivatives and Homogeneity: Euler's theorem provides a crucial link between partial derivatives and the homogeneity of a function.

  • Simplification Technique: It allows simplification of expressions involving homogeneous functions by relating partial derivatives and the function itself.