Jacobi determinant


Let

f=f(x)=f(x1,,xn)

be a function of n variables, and let

u=u(x)=(u1(x),,un(x))

be a function of x, where inversely x can be expressed as a function of u,

x=x(u)=(x1(u),,xn(u))

The formula for a change of variable in an n-dimensional integral is then

Ωf(x)dnx=u(Ω)f(x(u))|det(dx/du)|dnu

Ω is an integration region, and one integrates over all xΩ, or equivalently, all uu(Ω). dx/du=(du/dx)-1 is the Jacobi matrix and

|det(dx/du)|=|det(du/dx)|-1

is the absolute valuePlanetmathPlanetmath of the Jacobi determinant or Jacobian.

As an example, take n=2 and

Ω={(x1,x2)|0<x11,0<x21}

Define

ρ=-2log(x1)φ=2πx2u1=ρcosφu2=ρsinφ

Then by the chain rule and definition of the Jacobi matrix,

du/dx = (u1,u2)/(x1,x2)
= ((u1,u2)/(ρ,φ))((ρ,φ)/(x1,x2))
= (cosφ-ρsinφsinφρcosφ)(-1/ρx1002ϕ)

The Jacobi determinant is

det(du/dx) = det{(u1,u2)/(ρ,φ)}det{(ρ,φ)/(x1,x2)}
= ρ(-2π/ρx1)=-2π/xi

and

d2x = |det(dx/du)|d2u=|det(du/dx)|-1d2u
= (x1/2π)=(1/2π)exp(-(u12+u22/2))d2u

This shows that if x1 and x2 are independentPlanetmathPlanetmath random variablesMathworldPlanetmath with uniform distributionsMathworldPlanetmath between 0 and 1, then u1 and u2 as defined above are independent random variables with standard normal distributionsMathworldPlanetmath.

References

  • Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)

Title Jacobi determinant
Canonical name JacobiDeterminant
Date of creation 2013-03-22 12:07:08
Last modified on 2013-03-22 12:07:08
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 8
Author akrowne (2)
Entry type Definition
Classification msc 62H05
Classification msc 15-00
Synonym Jacobian
Related topic ChainRuleSeveralVariables
Related topic MultidimensionalGaussianIntegral
Related topic ChangeOfVariablesInIntegralOnMathbbRn