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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?

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Similar answer, a friend recently asked about twisted forms for non-orientable manifolds, what I found was pages 79-88 in Bott and Tu, "Differential forms in algebraic topology." I also think a twisted Stokes' Theorem is possible, as they present an entire twisted de Rham complex. Anyway, take a look:

I googled "twisted Stokes theorem." My friend Dmitry asked originally based on some physics inquiries. It appears that these physics people give a pretty direct discussion, maybe it is enough. "Foundations of classical electrodynamics: charge, flux, and metric" By Friedrich W. Hehl, Yuri N. Obukhov, (2003) Birkhauser

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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.

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A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).

EDIT: As explained in the Bott--Tu book (see the link in Will's answer and also these two pages), rescuing the Stokes theorem in the non-orientable case requires passing from differential forms to densities.

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