# Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor.

## Formal definition

Suppose that ${\displaystyle f:V\rightarrow W\qquad \qquad }$ is a function between two vector spaces over a field ${\displaystyle F\qquad \qquad }$.

We say that ${\displaystyle f\qquad \qquad }$ is homogeneous of degree ${\displaystyle k\qquad \qquad }$ if

${\displaystyle f(\alpha \mathbf {v} )=\alpha ^{k}f(\mathbf {v} )}$

for all nonzero ${\displaystyle \alpha \in F\qquad \qquad }$ and ${\displaystyle \mathbf {v} \in V\qquad \qquad }$.

## Examples

• A linear function ${\displaystyle f:V\rightarrow W\qquad \qquad }$ is homogeneous of degree 1, since by the definition of linearity
${\displaystyle f(\alpha \mathbf {v} )=\alpha f(\mathbf {v} )}$

for all ${\displaystyle \alpha \in F\qquad \qquad }$ and ${\displaystyle \mathbf {v} \in V\qquad \qquad }$.

• A multilinear function ${\displaystyle f:V_{1}\times \ldots \times V_{n}\rightarrow W\qquad \qquad }$ is homogeneous of degree n, since by the definition of multilinearity
${\displaystyle f(\alpha \mathbf {v} _{1},\ldots ,\alpha \mathbf {v} _{n})=\alpha ^{n}f(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})}$

for all ${\displaystyle \alpha \in F\qquad \qquad }$ and ${\displaystyle \mathbf {v} _{1}\in V_{1},\ldots ,\mathbf {v} _{n}\in V_{n}\qquad \qquad }$.

• It follows from the previous example that the ${\displaystyle n}$th Fréchet derivative of a function ${\displaystyle f:X\rightarrow Y}$ between two Banach spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is homogeneous of degree ${\displaystyle n}$.
• Monomials in ${\displaystyle n}$ real variables define homogeneous functions ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$. For example,
${\displaystyle f(x,y,z)=x^{5}y^{2}z^{3}}$

is homogeneous of degree 10 since

${\displaystyle (\alpha x)^{5}(\alpha y)^{2}(\alpha z)^{3}=\alpha ^{10}x^{5}y^{2}z^{3}}$.
${\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}}$

is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

## Elementary theorems

• Euler's theorem: Suppose that the function ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ is differentiable and homogeneous of degree ${\displaystyle k}$. Then
${\displaystyle \mathbf {x} \cdot \nabla f(\mathbf {x} )=kf(\mathbf {x} )\qquad \qquad }$.

This result is proved as follows. Writing ${\displaystyle f=f(x_{1},\ldots ,x_{n})}$ and differentiating the equation

${\displaystyle f(\alpha \mathbf {y} )=\alpha ^{k}f(\mathbf {y} )}$

with respect to ${\displaystyle \alpha }$, we find by the chain rule that

${\displaystyle {\frac {\partial }{\partial x_{1}}}f(\alpha \mathbf {y} ){\frac {\mathrm {d} }{\mathrm {d} \alpha }}(\alpha y_{1})+\cdots {\frac {\partial }{\partial x_{n}}}f(\alpha \mathbf {y} ){\frac {\mathrm {d} }{\mathrm {d} \alpha }}(\alpha y_{n})=k\alpha ^{k-1}f(\mathbf {y} )}$,

so that

${\displaystyle y_{1}{\frac {\partial }{\partial x_{1}}}f(\alpha \mathbf {y} )+\cdots y_{n}{\frac {\partial }{\partial x_{n}}}f(\alpha \mathbf {y} )=k\alpha ^{k-1}f(\mathbf {y} )}$.

The above equation can be written in the del notation as

${\displaystyle \mathbf {y} \cdot \nabla f(\alpha \mathbf {y} )=k\alpha ^{k-1}f(\mathbf {y} ),\qquad \qquad \nabla =({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}})}$,

from which the stated result is obtained by setting ${\displaystyle \alpha =1}$.

• Suppose that ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ is differentiable and homogeneous of degree ${\displaystyle k}$. Then its first-order partial derivatives ${\displaystyle \partial f/\partial x_{i}}$ are homogeneous of degree ${\displaystyle k-1\qquad \qquad }$.

This result is proved in the same way as Euler's theorem. Writing ${\displaystyle f=f(x_{1},\ldots ,x_{n})}$ and differentiating the equation

${\displaystyle f(\alpha \mathbf {y} )=\alpha ^{k}f(\mathbf {y} )}$

with respect to ${\displaystyle y_{i}}$, we find by the chain rule that

${\displaystyle {\frac {\partial }{\partial x_{i}}}f(\alpha \mathbf {y} ){\frac {\mathrm {d} }{\mathrm {d} y_{i}}}(\alpha y_{i})=\alpha ^{k}{\frac {\partial }{\partial x_{i}}}f(\mathbf {y} ){\frac {\mathrm {d} }{\mathrm {d} y_{i}}}(y_{i})}$,

so that

${\displaystyle \alpha {\frac {\partial }{\partial x_{i}}}f(\alpha \mathbf {y} )=\alpha ^{k}f(\mathbf {y} )}$

and hence

${\displaystyle {\frac {\partial }{\partial x_{i}}}f(\alpha \mathbf {y} )=\alpha ^{k-1}f(\mathbf {y} )}$.

## Application to ODEs

The substitution ${\displaystyle v=y/x}$ converts the ordinary differential equation

${\displaystyle I(x,y){\frac {\mathrm {d} y}{\mathrm {d} x}}+J(x,y)=0,}$

where ${\displaystyle I}$ and ${\displaystyle J}$ are homogeneous functions of the same degree, into the separable differential equation

${\displaystyle x{\frac {\mathrm {d} v}{\mathrm {d} x}}=-{\frac {J(1,v)}{I(1,v)}}-v}$.

## References

• Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. pp. p. 188. ISBN 3-540-09484-9.