# Homogeneous function

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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$f: V arr W qquadqquad$is a function between two vector spaces over a field $F qquadqquad$.

We say that $f qquadqquad$ is "homogeneous of degree $k qquadqquad$" if :$f\left(alpha mathbf\left\{v\right\}\right) = alpha^k f\left(mathbf\left\{v\right\}\right)$for all nonzero $alpha isin F qquadqquad$ and $mathbf\left\{v\right\} isin V qquadqquad$.

Examples

*A linear function $f: V arr W qquadqquad$ is homogeneous of degree 1, since by the definition of linearity:$f\left(alpha mathbf\left\{v\right\}\right)=alpha f\left(mathbf\left\{v\right\}\right)$for all $alpha isin F qquadqquad$ and $mathbf\left\{v\right\} isin V qquadqquad$.

*A multilinear function $f: V_1 imes ldots imes V_n arr W qquadqquad$ is homogeneous of degree n, since by the definition of multilinearity:$f\left(alpha mathbf\left\{v\right\}_1,ldots,alpha mathbf\left\{v\right\}_n\right)=alpha^n f\left(mathbf\left\{v\right\}_1,ldots, mathbf\left\{v\right\}_n\right)$for all $alpha isin F qquadqquad$ and $mathbf\left\{v\right\}_1 isin V_1,ldots,mathbf\left\{v\right\}_n isin V_n qquadqquad$.

*It follows from the previous example that the $n$th Fréchet derivative of a function $f: X ightarrow Y$ between two Banach spaces $X$ and $Y$ is homogeneous of degree $n$.

*Monomials in $n$ real variables define homogeneous functions $f:mathbb\left\{R\right\}^n arr mathbb\left\{R\right\}$. For example,:$f\left(x,y,z\right)=x^5y^2z^3$is homogeneous of degree 10 since:$\left(alpha x\right)^5\left(alpha y\right)^2\left(alpha z\right)^3=alpha^\left\{10\right\}x^5y^2z^3$.

*A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example, :$x^5 + 2 x^3 y^2 + 9 x y^4$is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneousfunctions.

Elementary theorems

*Euler's theorem: Suppose that the function $f:mathbb\left\{R\right\}^n arr mathbb\left\{R\right\}$ is differentiable and homogeneous of degree $k$. Then:$mathbf\left\{x\right\} cdot abla f\left(mathbf\left\{x\right\}\right)= kf\left(mathbf\left\{x\right\}\right) qquadqquad$.

This result is proved as follows. Writing $f=f\left(x_1,ldots,x_n\right)$ and differentiating the equation:$f\left(alpha mathbf\left\{y\right\}\right)=alpha^k f\left(mathbf\left\{y\right\}\right)$with respect to $alpha$, we find by the chain rule that:$frac\left\{partial\right\}\left\{partial x_1\right\}f\left(alphamathbf\left\{y\right\}\right)frac\left\{mathrm\left\{d\left\{mathrm\left\{d\right\}alpha\right\}\left(alpha y_1\right)+ cdotsfrac\left\{partial\right\}\left\{partial x_n\right\}f\left(alphamathbf\left\{y\right\}\right)frac\left\{mathrm\left\{d\left\{mathrm\left\{d\right\}alpha\right\}\left(alpha y_n\right) = k alpha ^\left\{k-1\right\} f\left(mathbf\left\{y\right\}\right)$,so that:$y_1frac\left\{partial\right\}\left\{partial x_1\right\}f\left(alphamathbf\left\{y\right\}\right)+ cdotsy_nfrac\left\{partial\right\}\left\{partial x_n\right\}f\left(alphamathbf\left\{y\right\}\right) = k alpha^\left\{k-1\right\} f\left(mathbf\left\{y\right\}\right)$.The above equation can be written in the del notation as:$mathbf\left\{y\right\} cdot abla f\left(alpha mathbf\left\{y\right\}\right) = k alpha^\left\{k-1\right\}f\left(mathbf\left\{y\right\}\right), qquadqquad abla=\left(frac\left\{partial\right\}\left\{partial x_1\right\},ldots,frac\left\{partial\right\}\left\{partial x_n\right\}\right)$,from which the stated result is obtained by setting $alpha=1$.

*Suppose that $f:mathbb\left\{R\right\}^n arr mathbb\left\{R\right\}$ is differentiable and homogeneous of degree $k$. Then its first-order partial derivatives $partial f/partial x_i$ are homogeneous of degree $k-1 qquadqquad$.

This result is proved in the same way as Euler's theorem. Writing $f=f\left(x_1,ldots,x_n\right)$ and differentiating the equation:$f\left(alpha mathbf\left\{y\right\}\right)=alpha^k f\left(mathbf\left\{y\right\}\right)$with respect to $y_i$, we find by the chain rule that:$frac\left\{partial\right\}\left\{partial x_i\right\}f\left(alphamathbf\left\{y\right\}\right)frac\left\{mathrm\left\{d\left\{mathrm\left\{d\right\}y_i\right\}\left(alpha y_i\right) = alpha ^k frac\left\{partial\right\}\left\{partial x_i\right\}f\left(mathbf\left\{y\right\}\right)frac\left\{mathrm\left\{d\left\{mathrm\left\{d\right\}y_i\right\}\left(y_i\right)$,so that:$alphafrac\left\{partial\right\}\left\{partial x_i\right\}f\left(alphamathbf\left\{y\right\}\right) = alpha ^k frac\left\{partial\right\}\left\{partial x_i\right\}f\left(mathbf\left\{y\right\}\right)$and hence:$frac\left\{partial\right\}\left\{partial x_i\right\}f\left(alphamathbf\left\{y\right\}\right) = alpha ^\left\{k-1\right\} frac\left\{partial\right\}\left\{partial x_i\right\}f\left(mathbf\left\{y\right\}\right)$.

Application to ODEs

The substitution $v=y/x$ converts the ordinary differential equation: $I\left(x, y\right)frac\left\{mathrm\left\{d\right\}y\right\}\left\{mathrm\left\{d\right\}x\right\} + J\left(x,y\right) = 0,$where $I$ and $J$ are homogeneous functions of the same degree, into the separable differential equation:$x frac\left\{mathrm\left\{d\right\}v\right\}\left\{mathrm\left\{d\right\}x\right\}=-frac\left\{J\left(1,v\right)\right\}\left\{I\left(1,v\right)\right\}-v$.

References

*cite book | author=Blatter, Christian | title=Analysis II (2nd ed.) | publisher=Springer Verlag | year=1979 |language=German |isbn=3-540-09484-9 | pages=p. 188 | chapter=20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.

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