Problem with Stokes' theorem

Question

I have read somewhere that the the integral of the exterior derivative of a one form gives zero by Stokes’ theorem, e.g.,

$$ \int_M d(f_i dx^i) =0 \qquad\text{for} \qquad f_i\in C^\infty(M)\,. $$

But why?

Answer 1

Based on the form you are integrating, I assume $M$ is a 2-manifold. By Stokes' theorem we get

$$ \int_M d(f_i \, dx^i) = \int_{\partial M} f_i \, dx^i. $$

Now, what you have probably seen is that the last expression evaluates to zero when $M$ is a manifold without boundary, i.e., $\partial M = \varnothing$. This holds in general for any integral of an exact form over a n-manifold without boundary.