Homogeneous function


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(alpha mathbf{v}) = alpha^k f(mathbf{v}) for all nonzero alpha isin F qquadqquad and mathbf{v} isin V qquadqquad.

Examples

*A linear function f: V arr W qquadqquad is homogeneous of degree 1, since by the definition of linearity:f(alpha mathbf{v})=alpha f(mathbf{v})for all alpha isin F qquadqquad and mathbf{v} 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(alpha mathbf{v}_1,ldots,alpha mathbf{v}_n)=alpha^n f(mathbf{v}_1,ldots, mathbf{v}_n)for all alpha isin F qquadqquad and mathbf{v}_1 isin V_1,ldots,mathbf{v}_n isin V_n qquadqquad.

*It follows from the previous example that the nth 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{R}^n arr mathbb{R}. For example,:f(x,y,z)=x^5y^2z^3is homogeneous of degree 10 since:(alpha x)^5(alpha y)^2(alpha z)^3=alpha^{10}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^4is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneousfunctions.

Elementary theorems

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

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

*Suppose that f:mathbb{R}^n arr mathbb{R} 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(x_1,ldots,x_n) and differentiating the equation:f(alpha mathbf{y})=alpha^k f(mathbf{y})with respect to y_i, we find by the chain rule that:frac{partial}{partial x_i}f(alphamathbf{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:alphafrac{partial}{partial x_i}f(alphamathbf{y}) = alpha ^k frac{partial}{partial x_i}f(mathbf{y})and hence:frac{partial}{partial x_i}f(alphamathbf{y}) = alpha ^{k-1} frac{partial}{partial x_i}f(mathbf{y}).

Application to ODEs

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

External links

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