Question

Explain eulars theorem

Answer 1

Euler's theorem states that if $f(x_1, x_2, ..., x_k)$ is a homogeneous function of degree $n$ with continuous first-order partial derivatives, then $\sum_{i=1}^k x_i \frac{\partial f}{\partial x_i} = nf$.

Answer 2

Euler's theorem for homogeneous functions relates the partial derivatives of a homogeneous function to the function itself and its degree.

Definition of a Homogeneous Function:

A function $f(x_1, x_2, ..., x_k)$ is said to be homogeneous of degree $n$ if for any non-zero scalar $t$,

$$f(tx_1, tx_2, ..., tx_k) = t^n f(x_1, x_2, ..., x_k).$$

For example, a function $f(x, y) = x^2 + 3xy + y^2$ is homogeneous of degree 2 because $f(tx, ty) = (tx)^2 + 3(tx)(ty) + (ty)^2 = t^2x^2 + 3t^2xy + t^2y^2 = t^2(x^2 + 3xy + y^2) = t^2 f(x, y)$.

A function $g(x, y) = x^3 + y^2$ is not homogeneous because $g(tx, ty) = (tx)^3 + (ty)^2 = t^3x^3 + t^2y^2$, which cannot be written in the form $t^n (x^3 + y^2)$.

Statement of Euler's Theorem:

If $f(x_1, x_2, ..., x_k)$ is a homogeneous function of degree $n$ with continuous first-order partial derivatives, then the following relationship holds:

$$x_1 \frac{\partial f}{\partial x_1} + x_2 \frac{\partial f}{\partial x_2} + ... + x_k \frac{\partial f}{\partial x_k} = nf.$$

For a function of two variables, $f(x, y)$, homogeneous of degree $n$:

$$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = nf.$$

For a function of three variables, $f(x, y, z)$, homogeneous of degree $n$:

$$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = nf.$$

This theorem is a fundamental result in the study of homogeneous functions and has applications in various fields, including economics (e.g., production functions) and differential equations.