New Problem: Transformation of Volumes

Here’s a real life problem that came up when I was looking for dark matter (I checked under my bed like six times, but I didn’t find any.  When I presented this research at group meeting, everyone got really quiet.  I think they were in awe.)

In the problem below, I use “volume” where you might think “area” makes more sense.  2D volume is area, so same thing.  But I use the term “volume” to suggest that you might want to generalize the problem to higher dimensions.

Suppose you have some 2-D volume in the plane.  You then do some sort of pointwise transformation, which maps this into a new volume, like so:

A super-spooky example of a smooth pointwise transformation from R2 to R2

You can describe this transformation by

where is the coordinate to which you map the point .  Assume that and are smooth and invertible.  Points inside the region stay inside the region, and things behave nicely, if weirdly.  Straight lines map onto curvy lines (or straight lines, if they want), but not broken up segments or points.  The neighborhood of a point transforms into the neighborhood of the transformed point.  Different points stay different (never wind up on top of each other).  Photographs of your head become distorted, but your eye (if you have one, this is an equal-opportunity blog) is still next to your nose.

Question: What is the area of the transformed region, in terms of the old region and the transformation equations?  (see tags for a hint or two)

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Tags: area, change of variables, hint: jacobian, hint: transform a very small square first, math, multivariable calculus, transformation, volume, volume element