In these course notes, the author constructs a $1$-form from a given homology generator in the following way: 
Then, he claims that the integral of $\omega$ over the generator is nonzero, and hints at using Stokes' Theorem. I'm struggling to see why this is the case, it would seem to me that since the generator $l$ is a closed loop, this integral should be zero, since every time I enter then exit a triangle, the total integral is $0$. What am I missing?
Edit: I have been suspecting that the notes are wrong, and the author intended to convey that what has a nonzero integral is not the $1$-form $\omega$ but rather its harmonic part in the Helmholtz-Hodge decomposition. I'll close the question for now