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Let $R \subset \mathbb{R}^2$ and $R' \subset \mathbb{R}^2$ be two regions in the plane, and $F: R \rightarrow R'$ a smooth map.

I would like to find a reasonable measure of the "regularity" (not in any technical sense, just "little distortion") of $F$, like in the following image:

good/bad mapping

The original region is on the left, and its images under two mappings are displayed to the right, one of them "good", the other one "bad" (i.e. highly distorted).

My question: how can I put a number to "good" vs. "bad"? I've thought about integrating the Laplacian of $F$, $\int_R \Delta F\ dx$, but it is defined only for scalar-valued functions $f:\mathbb{R}^2 \rightarrow \mathbb{R}$. Maybe something related?

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It seems to me like you want to measure the curvature of this mapping in some sense. What about integrating the scalar curvature induced by the mapping $F:R\to R'$? –  Jeff Nov 8 '11 at 20:56

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