Another question on Dirac's GR book

[Mike_Fontenot]

I'm stuck again on a statement Dirac makes in chapter 26 of his GR book,
on the gravitational action principle (p. 48).

He has the integral

I = {int} R sqrt(-g) d{sup 4}x,

where R is the scalar curvature (the contraction of the Ricci tensor), g
is the determinant of the metric, and

d{sup 4}x = dx{sup 0} dx{sup 1} dx{sup 2} dx{sup 3},

i.e., a differential element of 4-dimensional volume.

He then writes R as the sum of a bunch of terms, and says that the first
two of those terms "are perfect differentials, so they will contribute
nothing to I".

The first of those two terms is

( g{sup mu nu} gamma{sup sigma}sub{mu sigma} sqrt(-g) ){sub ,nu}

where the comma in the final subscript denotes the partial derivative
wrt x{sup nu}.

I can see how he GETS that first term, but I can't see why it
contributes nothing to the integral I. I don't understand what he means
by "a perfect differential" in this case, although I think I normally
understand that terminology OK. He also says that he is going to make
use of "partial integration", and I don't see where he's using that in
the above. I've used integration by parts before in numerous
situations, but I don't see how he's using it in this case.

I suspect that some of my problem may be that, although I've used
integration by parts many times before, I may not understand how it
generalizes to 4 dimensions, and to functions of more than one variable
(so that the ordinary derivatives are replaced by partial derivatives,
and the integration is over a 4-D volume rather than along a single
dimension).

Can anyone give me some tips on what he's doing here?

Mike Fontenot