# Euler’s theorem on homogeneous functions

###### Theorem 1 (Euler).

Let $f(x_{1},\ldots,x_{k})$ be a smooth homogeneous function of degree $n$. That is,

 $f(tx_{1},\ldots,tx_{k})=t^{n}f(x_{1},\ldots,x_{k}).{}$ (*)

Then the following identity holds

 $x_{1}\frac{\partial f}{\partial x_{1}}+\cdots+x_{k}\frac{\partial f}{\partial x% _{k}}=nf.$
###### Proof.

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

 $x_{1}\frac{\partial f}{\partial x_{1}}(tx_{1},\ldots,tx_{k})+\cdots+x_{k}\frac% {\partial f}{\partial x_{k}}(tx_{1},\ldots,tx_{k})=nt^{n-1}f(x_{1},\ldots,x_{k% }).$

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

Sometimes the differential operator $\displaystyle{x_{1}\frac{\partial}{\partial x_{1}}+\cdots+x_{k}\frac{\partial}% {\partial x_{k}}}$ is called the Euler operator. An equivalent way to state the theorem is to say that homogeneous functions are eigenfunctions of the Euler operator, with the degree of homogeneity as the eigenvalue.

Title Euler’s theorem on homogeneous functions EulersTheoremOnHomogeneousFunctions 2013-03-22 15:18:58 2013-03-22 15:18:58 CWoo (3771) CWoo (3771) 10 CWoo (3771) Theorem msc 26B12 msc 26A06 msc 15-00 Euler operator