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In a 1-1 transformation, the Jacobian term depends solely on the form of the transformation, and not the range?

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When computing the Jacobian, we need to apply an absolute value at the end, and the result of this may be affected by the ranges.

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For example, if our Jacobian term is $|v|$, this would simply become $v$ if $v>0$.

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Otherwise, if $v$ is unbounded, then $|v| = v$ when $v>0$ and $|v| = -v$ when $v<0$.

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