in my GR book, it claims that integral of a covariant divergence reduces to a surface term. I'm not sure if I see this...(adsbygoogle = window.adsbygoogle || []).push({});

So, is it true that:

[tex]\int_{\Sigma}\sqrt{-g}\nabla_{\mu} V^{\mu} d^nx= \int_{\partial\Sigma}\sqrt{-g} V^{\mu} d^{n-1}x[/tex]

if so, how do I make sense of the [itex]d^{n-1}x[/itex] term? would it be just a differential form?? so what about the following?

[tex]\int_{\Sigma} \nabla_{\mu} V^{\mu} d\omega=\int_{\partial \Sigma} \sqrt{-g} V^{\mu} \omega[/tex]

is it true? It certainly doesn't seem so to me... since the proof of stokes' theorem heavily rely on the similarities between taking the boundary of a simplex and taking the derivative of a form. This certainly does not seem to be the case with covariant derivatives. Perhaps it is only true with taking divergences. How do I go by proving/reasoning it? (simply saying things in flat space generalize in things in curved space with ; replaced by , doesn't do it for me)

edit: i just can't figure out what the heck is wrong with the latex... someone might have to fix it for me. (something is clearly messed up about the current latex system... seriously how can my d^{n-1} x term have any syntax error?

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# Stokes theorem under covariant derivaties?

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