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Can coordinate description cause numerical integrals to vary?

Luther Nayhm
Posted 11 years ago

The title does not quite capture the issue. I have a function I am integrating over a spherical volume, actually two spherical volumes. I define a vector that originated within one volume and terminates in the second volume. I now integrate this vector function over both volumes.

When I set up the integrand in spherical coordinates vs using cylindrical coordinate...by symmetry it is easy to use either coordinate system...the numerical integration does not give the same results. There is a persistent "bias" in the spherical coordinate system. It is easy to determine which approach is in error since there is a limit each integral should parametrically approach.

If I have in fact not made an error in the integrand in one or the other coordinate system, is this outcome a possibility using numerical integation? The integrals are not analytical.

I can get into more details but first I would like to know if there is something "quirky" in Mathematica relating to such an issue.
POSTED BY: Luther Nayhm
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Luther Nayhm
Luther Nayhm
Posted 11 years ago

I think I found an error.

If you would, could we take this part of the discussion off line, because it does not add to addressing my initial question.

Once we get past this issue we can resume the discussion here.

Luther@LutherNayhm.org
POSTED BY: Luther Nayhm
Luther Nayhm
Luther Nayhm
Posted 11 years ago

Interesting approach. Much more compact than my own, which is simply klunky lumping everything into one statement.

I will play with your code and see if I find how or why my calculations are so divergent.

This took a bit of effort on your part and I appreciate your efforts. I am unfamiliar with some of you syntax but I will follow up with a postmortem.

When I executed your equivalence statements, the first one simply ran until I aborted and then did it again, at which point it executed instantly. I have had that issue before in defining some function. I also notice that when I copy and paste, sometimes I have to delete and reenter certain code for it to execute. Not sure why it is happening.

But thanks again for your efforts.

POSTED BY: Luther Nayhm

I understand you want to calculate the interaction between the two spheres. If I do it I do not find a difference between using polar or cylindrical coordinates for sphere2. See notebook. Ok., there is a small difference, but that is certainely due to numerical errors in the integration.

regards hd

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