Jacobi determinant

Let

f=f(x)=f(x_{1},\ldots,x_{n})

be a function of $n$ variables, and let

u=u(x)=(u_{1}(x),\ldots,u_{n}(x))

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

x=x(u)=(x_{1}(u),\ldots,x_{n}(u))

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

\int_{\Omega}f(x)d^{n}x=\int_{u(\Omega)}f(x(u))|\det(dx/du)|d^{n}u

$\Omega$ is an integration region, and one integrates over all $x\in\Omega$ , or equivalently, all $u\in u(\Omega)$ . $dx/du=(du/dx)^{-1}$ is the Jacobi matrix and

|\det(dx/du)|=|\det(du/dx)|^{-1}

is the absolute value of the Jacobi determinant or Jacobian.

As an example, take $n=2$ and

\Omega=\{(x_{1},x_{2})|0<x_{1}\leq 1,0<x_{2}\leq 1\}

Define

\begin{array}[]{cc}\rho=\sqrt{-2\log(x_{1})}&\varphi=2\pi x_{2}\\ u_{1}=\rho\cos\varphi&u_{2}=\rho\sin\varphi\end{array}

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

$\displaystyle du/dx$	$\displaystyle=$	$\displaystyle\partial(u_{1},u_{2})/\partial(x_{1},x_{2})$
	$\displaystyle=$	$\displaystyle(\partial(u_{1},u_{2})/\partial(\rho,\varphi))(\partial(\rho,% \varphi)/\partial(x_{1},x_{2}))$
	$\displaystyle=$	$\displaystyle\begin{pmatrix}\cos\varphi&-\rho\sin\varphi\\ \sin\varphi&\rho\cos\varphi\end{pmatrix}\begin{pmatrix}-1/\rho x_{1}&0\\ 0&2\phi\end{pmatrix}$

The Jacobi determinant is

	$\displaystyle\det(du/dx)$	$\displaystyle=$	$\displaystyle\det\{\partial(u_{1},u_{2})/\partial(\rho,\varphi)\}\det\{% \partial(\rho,\varphi)/\partial(x_{1},x_{2})\}$
		$\displaystyle=$	$\displaystyle\rho(-2\pi/\rho x_{1})=-2\pi/x_{i}$

and

	$\displaystyle d^{2}x$	$\displaystyle=$	$\displaystyle\|\det(dx/du)\|d^{2}u=\|\det(du/dx)\|^{-1}d^{2}u$
		$\displaystyle=$	$\displaystyle(x_{1}/2\pi)=(1/2\pi)\exp(-(u_{1}^{2}+u_{2}^{2}/2))d^{2}u$

This shows that if $x_{1}$ and $x_{2}$ are independent random variables with uniform distributions between 0 and 1, then $u_{1}$ and $u_{2}$ as defined above are independent random variables with standard normal distributions.

References

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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