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In the book "A Relativistic toolkit - Eric Poisson", proof of Gauss Stokes theorem is given.This is the Gauss-Stokes theorem

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In the proof they have tried to show that LHS=RHS in a particular cooridnate system. I haven't understood that how do we eliminate the second term in the second line. The explanation which is given in the book is that the integration is over a closed three dimensional surface and x^a are angular coordinates. How does this work?

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As suggested in the referenced book, Lets think in 1 dimension less - imagine foliating a ball with spheres, and $x_{0}$ parametrizes the radius.

The term which is claimed to vanish is $\int _{x_{0}=c}d^3x(\sqrt {-g}A^{a})_{,a}$. In our 2+1 dimensional picture, this is an integral over a sphere of a divergence. The sphere has no boundary, so by Stoke's theorem it is 0. The same thing happens in any dimension, if the $x_{0}=c$ solutions are 3 dimensional manifolds without boundary.

I recommend the appendix of Carroll's GR book for an explanation on Stokes and divergence theorem in GR.

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