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I found a good lecture about Jacobian Matrix which was part of a statistics course. However, it was published 20 years ago and lack of explanation. As a beginner of statistics, I'm not able to find the relation between $f_{XY}, f_{UV}$ and respective area.

Can anyone help me understand the highlighted text? I've attached a screenshot of the lecture in the following. You can also find the original lecture from the bottom link.

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https://faculty.math.illinois.edu/~r-ash/Stat/StatLec1-5.pdf

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    $\begingroup$ The algebra behind the Jacobian is explained in my post at stats.stackexchange.com/a/154298/919. The connection with "area" (actually, hypervolume in $n$ instead of $2$ dimensions) can also be established algebraically: it comes down to the fact there is, up to a scalar multiple, a unique $n$-form in $n$ dimensions. This is fairly obvious once you realize that any $n$-form in $dx_1, \ldots, dx_n$ must contain each of the $dx_i$ exactly once and that reordering them only changes the sign. Finally, observe that any signed hypervolume must also be an $n$-form, QED. $\endgroup$ – whuber Nov 16 '18 at 22:20

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