Suppose $f$ is a non-constant meromorphic function on a compact Riemann manifold and $a\in \mathbb{C}$. I'm trying to understand why $$\mathrm{res}_P\left(\frac{df}{f-a}\right) = \frac{1}{2\pi i }\int_\gamma \frac{df}{f-a} = \mathrm{ord}_P(f-a)$$ where $\gamma$ is a simple loop containing $P$ (contained in one coordinate chart).
The proof I'm reading then claims that since the sum of residues of a meromorphic 1-form is 0, we have $$ (\mbox{multiplicity of } f=a) = \sum_P \max(0,\mathrm{ord}_P f) $$ and I don't see how this follows.