Jacobi determinant

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Let

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be a function of n variables, and let

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

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

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Hepa img504.gif is an integration region, and one integrates over all Hepa img505.gif , or equivalently, all Hepa img506.gif . Hepa img507.gif is the Jacobi matrix and

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is the absolute value of the Jacobi determinant or Jacobian.

As an example, take n=2 and

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Define

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Then by the chain rule ( Hepa img2.gif Jacobi Matrix)

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The Jacobi determinant is

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and

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This shows that if x1 and x2 are independent random variables with uniform distributions between 0 and 1, then u1 and u2 as defined above are independent random variables with standard normal distributions ( Hepa img2.gif Transformation of Random Variables).