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: $f(tx,ty)={t}^{n}\cdot f(x,y)$ 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: $x\cdot \frac{\mathrm{\partial}f}{\mathrm{\partial}x}+y\cdot \frac{\mathrm{\partial}f}{\mathrm{\partial}y}=n\cdot f(x,y)$
Proof of Euler's Theorem:

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

Apply the chain rule to differentiate $f(tx,ty)$ with respect to $t$: $\frac{df(tx,ty)}{dt}=n\cdot {t}^{n1}\cdot f(x,y)$

Chain rule for partial derivatives: $\frac{\mathrm{\partial}f}{\mathrm{\partial}x}\cdot \frac{dx}{dt}+\frac{\mathrm{\partial}f}{\mathrm{\partial}y}\cdot \frac{dy}{dt}=n\cdot {t}^{n1}\cdot f(x,y)$

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