I have been trying to understand what exactly the Poincaré residue is in the context of differential forms on algebraic varieties. I see that they occur in other contexts as well.
Let $ f \in \mathbb{C}[z_{1},\dots,z_{n}]. $ If we have an open set $ U_{i} = \Big\lbrace \frac{\partial f}{\partial z_{n}} \neq 0 \Big\rbrace, $ of the hypersurface $ V = \lbrace f=0 \rbrace \subset \mathbb{A}^{n}_{\mathbb{C}}, $ then we have the form $$ \omega_{i} = (-1)^{i}\frac{1}{{\partial f}/{\partial z_{i}}}dz_{1} \wedge \dots \wedge \widehat{dz_{i}} \wedge \dots \wedge dz_{n} \in \Omega^{n-1}(U_{i}). $$
Is this form what the Poincaré residue is? What purposes does it serve?