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Jacobi determinant (Definition)

Let

$\displaystyle f=f(x) = f(x_1,\ldots,x_n) $

be a function of $ n$ variables, and let

$\displaystyle 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$,

$\displaystyle x = x(u) = (x_1(u), \ldots,x_n(u)) $

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

$\displaystyle \int_\Omega f(x)d^nx = \int_{u(\Omega)} f(x(u))\vert\det(dx/du)\vert 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

$\displaystyle \vert \det(dx/du)\vert = \vert\det(du/dx)\vert^{-1} $

is the absolute value of the Jacobi determinant or Jacobian.

As an example, take $ n=2$ and

$\displaystyle \Omega = \{(x_1,x_2) \vert 0<x_1 \le 1, 0 < x_2 \le 1 \} $

Define

$\displaystyle \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,


$\displaystyle du/dx$ $\displaystyle =$ $\displaystyle \partial(u_1,u_2) / \partial(x_1,x_2)$  
  $\displaystyle =$ $\displaystyle (\partial(u_1,u_2)/\partial(\rho,\varphi))(\partial(\rho,\varphi)/\partial(x_1,x_2))$  
  $\displaystyle =$ $\displaystyle \begin{pmatrix}\cos \varphi & - \rho \sin \varphi \\ \sin \varphi... ...rphi \end{pmatrix} \begin{pmatrix}-1 / \rho x_1 & 0 \\ 0 & 2 \phi \end{pmatrix}$  

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


$\displaystyle d^2 x$ $\displaystyle =$ $\displaystyle \vert \det(dx/du) \vert d^2u = \vert\det(du/dx)\vert^{-1} d^2 u$  
  $\displaystyle =$ $\displaystyle (x_1/2\pi) = (1/2 \pi) \exp(-(u_1^2+u_2^2/2))d^2 u$  

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



"Jacobi determinant" is owned by akrowne.
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See Also: chain rule (several variables), multidimensional Gaussian integral, change of variables in integral on $\mathbb{R}^n$

Other names:  Jacobian

Attachments:
change of variables in integral on $\mathbb{R}^n$ (Theorem) by stevecheng
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Cross-references: standard normal distributions, uniform distributions, random variables, independent, chain rule, absolute value, Jacobi matrix, region, integral, variables, function
There are 14 references to this entry.

This is version 4 of Jacobi determinant, born on 2002-01-04, modified 2007-02-14.
Object id is 1268, canonical name is JacobiDeterminant.
Accessed 13357 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 62H05 (Statistics :: Multivariate analysis :: Characterization and structure theory)
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