In textbooks Stokes' theorem is usually formulated for orientable manifolds (at least I couldn't find any version not using orientability). Is Stokes theorem: $\int\limits\_{M}d\omega=\int\limits\_{\partial M} \omega$ also true for non-orientable manifolds?
Are there any references, you can tell me?


Regards.



Stokes theorem for manifolds without orientation?
This is just more information on Jose's comment. 
Look at section 10 (pages 122 to 128) in:


Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008. [(pdf)](http://www.mat.univie.ac.at/~michor/dgbook.pdf)


On the orientable double cover $\tilde M$ of $M$ (given by $\lbrace \xi\in Or(M): |\xi|=1\rbrace$ using 10.7) there are two kinds differential forms, namely the eigenspaces for $\pm 1$ under the action of the covering map. These are the "formes paire" and "formes impaire" on $M$ in the book of de Rham; for these you formulate Stokes' theorem directly on $M$, and it is the same as the Stokes' theorem on $\tilde M$. Note that you also have the corresponding double cover of the boundary.


