= Repeated-point Abel differential formula
{title2=$d u_D(v)(\omega)=\sum_p\operatorname{res}_p(v_p\omega)$}
At a multiplicity-$r$ point, choose a local parameter and elementary symmetric coordinates $e_1,\ldots,e_r$ 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 $d(\sum z_i^k)=(-1)^{k-1}k\,d e_k$. 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 $r-1$.
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