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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 Jacobi determinant or Jacobian.
As an example, take $n=2$ and
$$ \Omega = \{(x_1,x_2) | 0<x_1 \le 1, 0 < x_2 \le 1 \} $$
Define
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.
References
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