Jacobi determinantJacobiDeterminant2002-01-04 22:13:302007-02-14 21:23:47Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}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
$$ \int_\Omega f(x)d^nx = \int_{u(\Omega)} f(x(u))|\det(dx/du)|d^nu $$
$\Omega$ 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 \emph{Jacobi determinant} or \emph{Jacobian}.
As an example, take $n=2$ and
$$ \Omega = \{(x_1,x_2) | 0<x_1 \le 1, 0 < x_2 \le 1 \} $$
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,
\begin{eqnarray*}
du/dx & = & \partial(u_1,u_2) / \partial(x_1,x_2) \\
& = & (\partial(u_1,u_2)/\partial(\rho,\varphi))(\partial(\rho,\varphi)/\partial(x_1,x_2)) \\
& = & \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}
\end{eqnarray*}
The Jacobi determinant is
\begin{eqnarray*}
\det (du/dx) & = & \det \{ \partial(u_1,u_2)/\partial(\rho,\varphi)\} \det\{\partial(\rho,\varphi) / \partial(x_1,x_2)\} \\
& = & \rho(-2 \pi /\rho x_1) = -2 \pi / x_i
\end{eqnarray*}
and
\begin{eqnarray*}
d^2 x & = & | \det(dx/du) | d^2u = |\det(du/dx)|^{-1} d^2 u \\
& = & (x_1/2\pi) = (1/2 \pi) \exp(-(u_1^2+u_2^2/2))d^2 u
\end{eqnarray*}
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.
{\bf References}
\begin{itemize}
\item Originally from The Data Analysis Briefbook
(\PMlinkexternal{http://rkb.home.cern.ch/rkb/titleA.html}{http://rkb.home.cern.ch/rkb/titleA.html})
\end{itemize}