  In 2000, Voiculescu proved an algebraic characterization of cyclic gradients
of noncommutative polynomials. We extend this remarkable result in two
different directions: first, we obtain an analogous characterization of free
gradients; second, we lift both of these results to Voiculescu's fundamental
framework of multivariable generalized difference quotient rings. For that
purpose, we develop the concept of divergence operators, for both free and
cyclic gradients, and study the associated (weak) grading and cyclic
symmetrization operators, respectively. One the one hand, this puts a new
complexion on the initial polynomial case, and on the other hand, it provides a
uniform framework within which also other examples - such as a discrete version
of the It\^o stochastic integral - can be treated.
