TITLE: Green's formula for nonorientable manifolds
QUESTION [8 upvotes]: Usually in differential geometry one proves the Stokes theorem and then obtains divergence theorem and Green's formulas as corollaries. However, divergence theorem is also valid for nonorientable riemannian manifolds when one replaces forms with densities. But then the Green's formulas should also be valid. Am I missing something? I haven't found a discussion about where precisely the orientability is needed. 
Further supposing Green formulas in nonorientable manifolds one could define variational formulation for example for elliptic PDEs there. I wonder if the (non)orientability has some effect on the solutions of such PDEs?

REPLY [4 votes]: So it seems that Bott & Tu are saying that orientability doesn't really matter: Stokes theorem remains valid if one replaces forms by densities. But then it's curious why this fact is not clearly stated in differential geometry books. For example some projective spaces are nonorientable and obviously projective spaces are fundamental in all mathematics. So it's funny why there is no explicit discussion of these matters in projective spaces.