Jacobi determinant

Let

$$f=f(x)=f(x_1,\mathrm{\dots },x_n)$$

be a function of $n$ variables, and let

$$u=u(x)=(u_1(x),\mathrm{\dots },u_n(x))$$

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

$$x=x(u)=(x_1(u),\mathrm{\dots },x_n(u))$$

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

$${\int }_{\mathrm{\Omega }}f(x)d^{n}x={\int }_{u(\mathrm{\Omega })}f(x(u))|det(dx/du)|d^{n}u$$

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

Define

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

$du/dx$$=$$\partial (u_1,u_2)/\partial (x_1,x_2)$

$=$$(\partial (u_1,u_2)/\partial (\rho ,\phi ))(\partial (\rho ,\phi )/\partial (x_1,x_2))$

$=$

The Jacobi determinant is

$det(du/dx)$$=$$det\{\partial (u_1,u_2)/\partial (\rho ,\phi )\}det\{\partial (\rho ,\phi )/\partial (x_1,x_2)\}$

$=$$\rho (-2\pi /\rho x_1)=-2\pi /x_i$

and

$d^{2}x$$=$$|det(dx/du)|d^{2}u={|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$

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)