This is my first course in differential forms so it might be a trivial question. If $\mu$ is a $n-1$-form on $n$-dim manifold $X$ the book uses that
$$\int_X{d\mu}=0.$$
Is this expression valid for all $\mu$ and $X$? If that is true doesn't by Stokes theorem follow that $\int_{\partial X}\mu=0$ for every $\mu$?
If $X$ is an $n$-dimensional manifold with boundary then for any $n-1$-form $\mu$ with compact support, $$\int_Xd\mu=\int_{\partial X}\mu.$$ This is Stokes' theorem. If there is no boundary, i.e., if $\partial X=\emptyset$, then the boundary integral is zero, and hence so is the other integral. But otherwise, you cannot expect the integral to vanish.