Homogeneous function

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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 is a function between two vector spaces over a field .

We say that is homogeneous of degree if

for all nonzero and .

Examples

  • A linear function is homogeneous of degree 1, since by the definition of linearity

for all and .

  • A multilinear function is homogeneous of degree n, since by the definition of multilinearity

for all and .

  • It follows from the previous example that the th Fréchet derivative of a function between two Banach spaces and is homogeneous of degree .
  • Monomials in real variables define homogeneous functions . For example,

is homogeneous of degree 10 since

.

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

Elementary theorems

  • Euler's theorem: Suppose that the function is differentiable and homogeneous of degree . Then
.

This result is proved as follows. Writing and differentiating the equation

with respect to , we find by the chain rule that

,

so that

.

The above equation can be written in the del notation as

,

from which the stated result is obtained by setting .

  • Suppose that is differentiable and homogeneous of degree . Then its first-order partial derivatives are homogeneous of degree .

This result is proved in the same way as Euler's theorem. Writing and differentiating the equation

with respect to , we find by the chain rule that

,

so that

and hence

.

Application to ODEs

The substitution converts the ordinary differential equation

where and are homogeneous functions of the same degree, into the separable differential equation

.

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.


External links

de:Homogene Funktion eo:Homogena funkcio he:פונקציה הומוגנית it:Funzione omogenea nl:Homogeniteit (analyse)


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