derivative of homogeneous function

Theorem 1.

Suppose f:RnRm is a differentiableMathworldPlanetmathPlanetmath positively homogeneous function of degree r. Then fxi is a positively homogeneous function of degree r-1.


By considering componentPlanetmathPlanetmath functions if necessary, we can assume that m=1. For λ, let Mλ be the multiplicationPlanetmathPlanetmath map,

Mλ:n n
v λv.

For λ>0 and vn, we have

fxi(λv) = (fMλM1/λ)xi(λv)
= l=1n(fMλ)xl(v)(xx/λ)lxi(λv)
= (fMλ)xi(v)1λ
= λr-1fxi(v)

as claimed. ∎

Title derivative of homogeneous function
Canonical name DerivativeOfHomogeneousFunction
Date of creation 2013-03-22 14:45:05
Last modified on 2013-03-22 14:45:05
Owner matte (1858)
Last modified by matte (1858)
Numerical id 9
Author matte (1858)
Entry type TheoremMathworldPlanetmath
Classification msc 15-00