Map jacobian determinant

I was thinking that there's some result about the integral of the map jacobian determinant. Here's some work

This bottom ? approximation is just wrong. Nothing I can do here unfortunately

When looking at the pullback of a map, there are two terms, namely

But what makes "look like" ? Consider two scenarios

1. , or possible, it has tails that are extremely heavy. Consider as the normal distribution and for some small . This will be basically for a large swath of time. On the other hand, will look like a Gaussian PDF (plus a constant)! In this setting, then, only really matters in the tails to ensure that the constant doesn't end up mattering all too much.
2. On the other hand, suppose . Then, basically, the transport map doesn't do anything. In some sense, this comes from a "divergence-free" transport. One can imagine this as a rotation (though rotations are only a tiny subclass of these). This will happen if , via a hand waving at the below