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There has to be a measure for the difference between "instantaneous" change of "energy" along a line in a space compared to averaged changed of energy along a line.

I could take a smooth surface in 3d (quadratic) and knock one point out of whack, and show the before vs. after in surface plots, but I think that is implicit in the question.

Is there a measure of fit that looks at some version of local "force by distance" versus a larger-scale integral of f dot-dr along the contour?

There is some measure of "heuristic" in the levels of line, or the grid spacing, but I suspect that when the question is reversed (inverted), if there is a decent answer, then the good metric can drive the selection of levels for level curves.

References or links would be appreciated.

There is an argument that the sum of forces to get back to an original point could be zero. I'm suspecting that the energy might not behave that way, and that the reference "larger scale" might be an input parameter like bin-count for a histogram. I'm pretty tired, so now is not the time.

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  • $\begingroup$ I'm having trouble interpreting most of this question--there are just too many ways to read it. Could you perhaps offer an example or an illustration of what you're after? $\endgroup$ – whuber Oct 23 '18 at 21:26
  • $\begingroup$ I hope to have some time this afternoon to work on it. I can ad a pair of pictures and say "these lines say something" and "how do I quantify the difference?". $\endgroup$ – EngrStudent Oct 24 '18 at 12:22

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