Euler’s theorem on homogeneous functions

Theorem 1 (Euler).

Let f(x1,,xk) be a smooth homogeneous function of degree n. That is,

f(tx1,,txk)=tnf(x1,,xk). (*)

Then the following identity holds


By homogeneity, the relationMathworldPlanetmath ((*)1) holds for all t. Taking the t-derivative of both sides, we establish that the following identity holds for all t:


To obtain the result of the theorem, it suffices to set t=1 in the previous formulaMathworldPlanetmathPlanetmath. ∎

Sometimes the differential operator x1x1++xkxk is called the Euler operator. An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath way to state the theorem is to say that homogeneous functions are eigenfunctions of the Euler operator, with the degree of homogeneity as the eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Title Euler’s theorem on homogeneous functions
Canonical name EulersTheoremOnHomogeneousFunctions
Date of creation 2013-03-22 15:18:58
Last modified on 2013-03-22 15:18:58
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Theorem
Classification msc 26B12
Classification msc 26A06
Classification msc 15-00
Defines Euler operator