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

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

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

Title Jacobi determinant JacobiDeterminant 2013-03-22 12:07:08 2013-03-22 12:07:08 akrowne (2) akrowne (2) 8 akrowne (2) Definition msc 62H05 msc 15-00 Jacobian ChainRuleSeveralVariables MultidimensionalGaussianIntegral ChangeOfVariablesInIntegralOnMathbbRn