The volume is preserved by the flow: where is the absolute value?

Question

Consider the following excerpt of the Liouville's theorem proof taken from "Arnold - mathematical methods of classical mechanics":

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In changing the variables in the integral, I don't understand why there is not the absolute value of the Jacobian determinant. Why the determinant of the Jacobian of the flow is positive?

Answer 1

Because $g^0$ is the identity, $g^t$ is a diffeomorphism, and $\det$ is continuous!

Answer 2

There isn't actually an absolute value in the change of variables formula. The absolute value only appears when you tweak the change of variables formula so that you flip the orientation of the region of integration whenever the Jacobian is negative.

Compare with ordinary one-dimensional definite integrals:

$$ \int_{g(a)}^{g(b)} f(x) \, dx = \int_a^b f(g(y)) g'(y) \, dy $$

As you can see, no absolute value: that's because the definite integral keeps track of the orientation of your interval.

If you instead think in terms of unoriented integrals, then you would need the absolute value:

$$ \int_{g([a,b])} f(x) \, dx = \int_{[a,b]} f(g(y)) |g'(y)| \, dy $$

to properly account for the possibility that $g(a) > g(b)$. (and this doesn't work at all if $g$ is not one-to-one on the domain $[a,b]$)

In higher dimensions, you have line integrals, surface integrals, and such which all do properly account for orientation, and thus you wouldn't need absolute values if you were using those.