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