Euler's Homogeneous Function Theorem

Posted on January 1, 2017.

While reading Landau and Lifschitz’s Mechanics book I came across a theorem that I had not encountered before. First, let us define a homogeneous function.

A function f is homogeneous of order k if f(tx1, …, txn) = tkf(x1, …, xn).

Euler’s Homogeneous Function Theorem then states

Let f(x1, …, xn) be a homogeneous function of order k. Then $$ k f(x_1, \dots, x_n) = \sum_{i=1}^n x_i \frac{\partial f}{\partial x_i}$$

Since f is homogeneous of order k, we have f(tx1, …, txn) = tkf(x1, …, xn). Taking the derivative with respect to t yields

$$ \sum_{i=1}^n \frac{\partial f}{\partial (tx_i)} \frac{\partial (tx_i)}{\partial t} = kt^{k-1} f(x_1, \dots, x_n) $$ $$ \sum_{i=1}^n x_i \frac{\partial f}{\partial (tx_i)} = kt^{k-1} f(x_1, \dots, x_n) $$

Taking t = 1 then yields

$$ \sum_{i=1}^n x_i \frac{\partial f}{\partial (x_i)} = k f(x_1, \dots, x_n) $$