Jacobi determinant
Let
be a function of variables, and let
be a function of , where inversely can be expressed as a function of ,
The formula for a change of variable in an -dimensional integral is then
is an integration region, and one integrates over all , or equivalently, all . is the Jacobi matrix and
is the absolute value of the Jacobi determinant or Jacobian.
As an example, take and
Define
Then by the chain rule and definition of the Jacobi matrix,
The Jacobi determinant is
and
This shows that if and are independent random variables with uniform distributions between 0 and 1, then and as defined above are independent random variables with standard normal distributions.
References
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Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
Title | Jacobi determinant |
---|---|
Canonical name | JacobiDeterminant |
Date of creation | 2013-03-22 12:07:08 |
Last modified on | 2013-03-22 12:07:08 |
Owner | akrowne (2) |
Last modified by | akrowne (2) |
Numerical id | 8 |
Author | akrowne (2) |
Entry type | Definition |
Classification | msc 62H05 |
Classification | msc 15-00 |
Synonym | Jacobian |
Related topic | ChainRuleSeveralVariables |
Related topic | MultidimensionalGaussianIntegral |
Related topic | ChangeOfVariablesInIntegralOnMathbbRn |