- Euler's theorem for homogeneous functions provides a relationship between a homogeneous function and its partial derivatives.
- A function is homogeneous of degree if it satisfies the following condition: where is a scalar and is a constant.
Euler's Theorem Statement:
- The theorem states that if is a homogeneous function of degree , then:
Proof of Euler's Theorem:
Start with the definition of a homogeneous function:
Apply the chain rule to differentiate with respect to :
Chain rule for partial derivatives:
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.
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.
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.