These integrals turn up in subjects such as quantum field theory. Yes, Divergence Theorem relates them but if I had a sphere in 3D space and took the surface integral, that wouldnt give me the volume, would it If not, what. So you can rewrite a surface integral to a volume integral and the other way round. So the surface has to be closed Otherwise the surface would not include a volume. The n + p = 0 mod 2 requirement is because the integral from −∞ to 0 contributes a factor of (−1) n+ p/2 to each term, while the integral from 0 to +∞ contributes a factor of 1/2 to each term. It relates the flux of a vector field through a surface to the divergence of vector field inside that volume.
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