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Mathematica

Integrate with assumption of symmetric volume

Danylo Malyuta, ETHZ
Posted 10 years ago

Hi! My problem is the following: I have the following matrix:

S={{0,-z,y},{z,0,-x},{-y,0,x}}}

The integral of this matrix over any symmetric domain is 0. However, the integral of S^2 is not zero over a symmetric domain. I do not know what this domain is! I want to do the following:

Integrate[f[S,S^2],x,y,z]

Where f[S,S^2] is a function I got from previous calculations that is very complicated and may have a bunch of S and S^2 terms here and there. I want Mathematica to do the simplification for me and eliminate automatically integrals of terms with just S in them, and keep integrals with S^2 terms.

I understand that this is what Mathematica is made for. I want to intervene as little as possible and make Mathematica do the simplification work - if I have to go through and manually set terms with just S to zero in a 10-line expression, the point of the software becomes useless. Please help me on this! I really appreciate it.
POSTED BY: Danylo Malyuta
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Hi, if your integrand is given in terms of S and S is not defined yet, then you could set the coefficient of S to zero:

expr = a S + b S^2 ;
expr = Collect[expr,S] /. Coefficient[expr,S] -> 0

or

expr = a S + b S^2 ;
expr = expr /. S -> S^n /. S^n -> 0 /. n -> 1

and after this define S explicitly and do the integrals.

I.M
POSTED BY: Ivan Morozov

I'm looking at another way. Suppose I have the big expression expr with many integrals. What if I do something like this:

Select[expr, MemberQ[#, S[x, y, z].S[x, y, z]] &]

My problem is the above does not work unless the integrand is exactly S[x, y, z].S[x, y, z]. How do I write the pattern such that the condition for True would be that just part of the integrand contains S[x, y, z].S[x, y, z]?

Thanks!
POSTED BY: Danylo Malyuta
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