# 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 $f\left(x,y\right)$ is homogeneous of degree $n$ if it satisfies the following condition: $f\left(tx,ty\right)={t}^{n}\cdot f\left(x,y\right)$ where $t$ is a scalar and $n$ is a constant.

### Euler's Theorem Statement:

• The theorem states that if $f\left(x,y\right)$ is a homogeneous function of degree $n$, then: $x\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }x}+y\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }y}=n\cdot f\left(x,y\right)$

### Proof of Euler's Theorem:

• Start with the definition of a homogeneous function: $f\left(tx,ty\right)={t}^{n}\cdot f\left(x,y\right)$

• Apply the chain rule to differentiate $f\left(tx,ty\right)$ with respect to $t$: $\frac{df\left(tx,ty\right)}{dt}=n\cdot {t}^{n-1}\cdot f\left(x,y\right)$

• Chain rule for partial derivatives: $\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\cdot \frac{dx}{dt}+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}\cdot \frac{dy}{dt}=n\cdot {t}^{n-1}\cdot f\left(x,y\right)$

• Substitute $t=1$: $x\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }x}+y\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }y}=n\cdot f\left(x,y\right)$

### 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.