ag.algebraic geometry – how to see the Gysin map explicitly in an easy situation

Let $C$ be a smooth projective curve and let $U subset C$ be an open affine subset, with closed complement $S$ consisting of a finite number of points. I am trying to see explicitly the Gysin map in algebraic de Rham cohomology
$$
H^0_{dR}(S) longrightarrow H^2_{dR}(C).
$$
By explicitly, I mean covering $C$ by $U$ and another affine open subset, say $V$, containing $S$ and computing $H^2_{dR}(C)$ a la Cech. In this way, every element can be represented by a global section of $Omega^1$ on $U cap V$. On the other hand, elements of $H^0_{dR}(S)$ are functions on $S$. So I am trying to find a way to produce a differential on $U cap V$ out of a function on $S$. I have the intuition that logarithmic derivative should play a role here but I am stuck. Can anyone help? Thanks!