My professor said that it's easy to generalize to these definitions of Stoke's and Gauss' theorem from the 3 dimensional versions but didn't say much else. He threw the following on the chalk board:
$$ Gauss' \\ \\ \int{A^{\mu} \frac{1}{6} \epsilon^{\mu \nu \rho \sigma} dS_{\nu \rho \sigma}} = \int{\partial_{\mu} A^{\mu} d{\Omega}} $$
$$ Stoke's \\ \\ \frac{1}{2} \int{A^{\mu \nu} \frac{1}{2} \epsilon_{\mu \nu \rho \sigma} dF^{\rho \sigma}} = \int{\frac{\partial A^{\mu \nu}} {\partial{x^{\nu}}} \frac{1}{6} \epsilon_{\mu \nu \rho \sigma} dS^{\mu \rho \sigma}}$$
Where $d\Omega = dx^0 dx^1 dx^2 dx^3$ is the invariant spacetime volume and $dS_{\mu \rho \sigma} $ $($ and $\ dS^{\mu \rho \sigma}), \ dF^{\rho \sigma} $ are 3-surface and 2-surface elements, respectively.
For Gauss' theorem, it makes sense why the 1/6 is present, for the normalization of the rank-4 levi-cevita symbol, but why is there only a factor of 1/4 for Stoke's theorem? Also, what is the role of the levi-cevita symbol in both theorems, I'm unsure how to interpret the 2-surface and 3-surface elements. Some resources for an easy example using both theorems in this notation would be great. Thanks!
