Free potential functions

TL;DR

This paper extends classical vector calculus theorems to free analysis, showing that free derivatives are curl-free and characterizing when a free vector field is a derivative of an analytic free map.

Contribution

It introduces free analogs of classical theorems, establishing conditions under which free derivatives are curl-free and correspond to derivatives of free analytic maps.

Findings

Free derivatives are necessarily curl-free.

On connected free domains, curl-free fields are derivatives of free analytic maps.

Provides a characterization of free derivatives in noncommutative settings.

Abstract

This article establishes free versions of two classical theorems: derivatives are curl-free and every curl-free vector field (on a simply connected domain) is a derivative. We show that the derivative of a noncommutative free analytic map must be free-curl free -- an analog of having zero curl. Moreover, under the assumption that the free domain is connected, this necessary condition is sufficient. Specifically, if $T$ is analytic free vector field defined on a connected free domain then $DT(X,H)[K,0] = DT(X,K)[H,0]$ if and only if there exists an analytic free map $f$ such that $Df(X)[H] = T(X,H)$.