Repeated-point Abel differential formula ($d{u}_D(v)(\omega )={\sum }_p{\mathrm{res}}_p(v_p\omega )$)

At a multiplicity-$r$ point, choose a local parameter and elementary symmetric coordinates $e_1,\dots ,e_r$ for the nearby roots. The tangent principal part is $v={\sum }_{k=1}^r(-1{)}^{k-1}\delta e_k{z}^{-k}$. Newton's identities give $d(\sum z_i^k)=(-1{)}^{k-1}kd{e}_k$. Expanding the primitive of a holomorphic one-form proves the residue formula, including its sign. The dual map is restriction to the length-$r$ scheme, so it sees all jets through order $r-1$.