Stokes' Theorem: line integrals around 2-faces of n-dimensional surface? Suppose we have a convex polytope in $n$ dimensions and are trying to calculate the surface integral (over this polytope) of some scalar function $f:R^n \rightarrow R$. Suppose all edges and vertices are known, so, in particular, all 2-faces are known.
Can this integral be decomposed via Stokes' theorem into a sum of line integrals tracing paths around the 2-faces? I know the orientation has to be gotten right for integrating each face, but is there anything else I am missing that stops me from "jumping" from $n$ dimensions down to 2?
Thanks in advance.
 A: Summarizing the extend discussion (sorry!) in the above comments,
You cannot apply Stoke's theorem repeatedly to reduce the dimension of the region of integration over and over, because the boundary of the boundary of set is empty.  For example, the boundary of the solid ball is the sphere, but the sphere has no boundary, so we have to stop there.
Moreover, Stokes theorem says (most generally) that $$\int_R dw = \int_{bR} w$$  So if you are trying to rewrite $\int_R v$ as an integral over $bR$ by using stokes theorem, you will at least need to know that there is some $w$ so that $v = dw$, i.e.  $v$ must be an exact form.
I recommend learning multivariable calculus well:  to me that means with differential forms.  Ted Shifrin's book on the subject is excellent.  I also plan on producing an online multivariable calculus class which will cover differential forms in the near future (That is part of why I am patrolling the multivariable calculus tag on this site!).  That will be available on gratisU.org, which is currently blank.
