Problem with integration of $1$-form on surface

Question

I have some problem with integration of differential forms on algebraic surfaces (I'm reading Cartan's book on analytic functions). Let $X \subseteq \mathbb{C}^2$ be an algebraic curve given by polynomial equation $z_1^3 + z_2^3 = 1$. Differentiating this equation we obtain $z_1^2 dz_1 + z_2^2 dz_2 = 0$. We define a form $\omega$ in neighborhood of any point with $z_2 \neq 0$ as $$ \omega = \frac{dz_1}{z_2}. $$ In neighborhood of points with $z_1 \neq 0$ we define $$ \omega = -\frac{z_2dz_2}{z_1^2}. $$ This form is defined correctly in the sense that definitions agree if both $z_1 \neq 0$ and $z_2 \neq 0$. I don't know how to integrate such form over some path on my surface, but I know how integrate forms over paths in $\mathbb{C}$. Please help me to understand integration over paths on surfaces within this example or give me some reference. (I did't find any example myself, only nude theory).

Answer 1

In general you break up the path into pieces which are contained in a chart, so you can integrate each piece in a specific chart (the same way you integrate forms over paths in $\mathbb{C}$), and then you add up the results from all the pieces.