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Homogeneous function

 

 
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Homogeneous function
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a homogeneous function is a function with multiplicative
Multiplicative

Multiplicative may refer to:*Multiplication*Multiplicative partition*A Multiplicative function...
 scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor.

ose that is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 between two vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
s over a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
 .

We say that is homogeneous of degree if for all nonzero and .

Linear functions Any linear
Linear

The word linear comes from the Latin word linearis, which means created by lines.In mathematics, a linear map or function f is a function which satisfies the following two properties......
 function is homogeneous of degree 1, since by the definition of linearity for all and .






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a homogeneous function is a function with multiplicative
Multiplicative

Multiplicative may refer to:*Multiplication*Multiplicative partition*A Multiplicative function...
 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
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 between two vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
s over a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
 .

We say that is homogeneous of degree if for all nonzero and .

Examples


Linear functions

Any linear
Linear

The word linear comes from the Latin word linearis, which means created by lines.In mathematics, a linear map or function f is a function which satisfies the following two properties......
 function is homogeneous of degree 1, since by the definition of linearity for all and . Similarly, any multilinear function is homogeneous of degree n, since by the definition of multilinearity for all and . It follows that the th Fréchet derivative
Fréchet derivative

In mathematics, the Fr?chet derivative is a derivative defined on Banach spaces. Named after Maurice Fr?chet, it is commonly used to formalize the concept of the functional derivative used widely in mathematical analysis, especially functional analysis....
 of a function between two Banach spaces and is homogeneous of degree .

Homogeneous polynomials

Monomials in real variables define homogeneous functions . For example, is homogeneous of degree 10 since . A homogeneous polynomial
Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree . For example, is a homogeneous polynomial...
 is a polynomial made up of a sum of monomials of the same degree. For example, is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Properties


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

This result is proved as follows. Writing and differentiating the equation with respect to , we find by the chain rule
Chain rule

In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...
 that , so that . The above equation can be written in the del
Del

In vector calculus, del is a vector differential operator represented by the nabla symbol: .Del is a mathematical tool serving primarily as a Convention for mathematical notation; it makes many equations easier to comprehend, write, and remember....
 notation as , from which the stated result is obtained by setting .

For the proof of the converse, see .

  • 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
Chain rule

In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...
 that , so that and hence .

Application to ODEs


The substitution converts the ordinary differential equation
Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable....
where and are homogeneous functions of the same degree, into the separable differential equation
Separable differential equation

In mathematics, a separable differential equation may refer to one of two related things, both of which are differential equations that can be attacked by a method of separation of variables....
 .

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