I'm looking at the deriviation of Einstein's equation via applying the principle of least action to the Hilbert-Einstein action. I'm trying to understand the vanishing of a term because it is a total derivative: http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec33.pdf, equation 19. My question is that, I've seen some sources that explain it via the generalized Stoke's theorem in differential forms which is a generalization of Gauss theorem, fundamental theorem of calculus etc. I'm wondering if, once the identity in equation 19 has been made, a covariant derivative expression being expressed as a solely partial derivative expression, can this term vanishing be justified via Gauss's theorem which states that the divergence of a vector field integrated over a volume is equal to the vector integrated over the corresponding surface of the volume? If the vector fields goes to zero fast enough at infinity. I'm not looking for rigor, just an argument that merely suffices. Thanks your help is greatly appreciated.