Free potential functions

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 $Df(X)[H]=T(X,H)$.

17 pages

References in corpus (4)

• Proper Analytic Free Maps
• Bianalytic free maps between spectrahedra and spectraballs
• The free Grothendieck theorem
• Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity

Cited by in corpus (1)

• Operator NC functions