Verifying angular moment integrals from Duderstadt and Hamilton

[bamaguy]

Show by direct calculation that Eqs. (4-134) and (4-137) in the textbook by Duderstadt and Hamilton hold, i.e.:(a) ∫ dΩΩiΩj= 4π/3 δij; i,j = x,y,z;
4π(b) ∫ dΩΩxΩyΩz = 0, if l, m, or n is odd.
4π

The integrals are over 4π.

This is part of the derivation of the diffusion equation from the neutron transport equation. Part (b) from D&H Next note that the integral of the product of any odd number of components of OMEGA vanishes by symmetry.

(a) I think that 4π/3 comes from the volume of the sphere and δij is the kronecker delta. I don't know how to show this mathematically.
(b) I think that this has to do with the sin or cos function.

 

[DEvens]

Not volume. Omega is solid angle.

http://en.wikipedia.org/wiki/Solid_angle

I don't have the textbook you cite. So it's a little difficult to follow the question. You should read back in the text to see if they don't do something on solid angle and how to manipulate it.