# Differential Form on a Riemann Surface

The following problem is basically from Miranda's "Algebraic Curves and Riemann Surfaces", which I am reading on my own; if there are any rules against posting textbook problems, my apologies!

Let $X$ be a smooth projective curve defined by the homogeneous polynomial $F(x,y,z)=0$, with $\deg F = d \geq 3$. Let $f(x,y) = F(x,y,1)$. Show that if $p(u,v)$ is a polynomial of degree at most $d-3$, then $p(u,v) \frac{du}{\partial f/ \partial v}$ defines a holomorphic 1-form on the compact Riemann surface X. If $X$ is not smooth, but has nodes, then this form is a holomorphic 1-form on the resolution.

I see that this is a holomorphic $1$-form on the affine curve defined by $f$, since the charts are just projection to the $x$ or $y$ coordinate; in the former case the form is evidently holomorphic and in the latter case the form transforms to $p(u,v) \frac{dv}{\partial f/ \partial u}$. However, I'm a bit confused as to the computations involved in checking this on the other affine curves, and what extra argument is needed for the nodes case.