Let compact riemannian surface X satisfy the equation $w^3=z(z-1)(z-2)$.
Let $p \in X$ be the point where $w=0, z=1$.
Compute
$Res_p(\frac{dz}{w^3}) $
By implicit function theorem, I see, that since $\frac{d(w^3 - z(z-1)(z-2)}{dz}$ is not zero at $p$, then the complement variable $w$ is a coordinate.
What does the differential form $dz$ mean if z is not a coordinate at $p$?
Please, give me a hint or a useful book about the problem.
You are absolutely right that at the point $p$ the variable $w$ is a local coordinate.
Let us make a slight change of coordinates and put $u=z-1$, so that $u(p)=w(p)=0$ and $X$ has equation $$w^3=u^3-u \quad (\ast)$$ Your differential form is then $\eta=\frac{dz}{w^3}=\frac{du}{w^3}$.
Since $(\ast)$ implies that $3w^2dw=(3u^2-1)du$ and thus that $du=\frac {3w^2dw}{3u^2-1}$, we see that $$\eta=\frac{du}{w^3}=\frac {3dw}{w(3u^2-1)}$$ Since $w$ is a local coordinate at $p$ and since $(3u^2-1)(p)=-1$ ,we obtain the desired result$$ \operatorname {Res}_p(\eta)=\frac {3}{-1}=-3$$