For functions of a single complex variable, points of multiplicity greater than $k$ are characterized by the vanishing of the first $k$ derivatives. There are various quantitative generalizations of this statement, showing that for functions that are in some sense close to having multiplicity greater than $k$, the first $k$ derivatives must be small. In this paper we aim to generalize this situation to the multi-dimensional setting. We define a class of differential operators, the \emph{multiplicity operators}, which act on maps from $\C^n$ to $\C^n$ and satisfy properties analogous to those described above. We demonstrate the usefulness of the construction by applying it to some problems in the theory of Noetherian functions.