Lipschitzian vector fields, upper gradients and distributional derivatives

We prove that given a locally integrable function fon an open set of an Euclidean space the distributional derivative Xfwith respect to a locally Lipschitzian vector field Xis locally integrable if, and only if, the function fadmits a locally integrable upper gradient along the vector field X; in this case Xfcoincides with the Lie derivative LXfand |Xf|is the least upper gradient of the function f. Applications to systems of locally Lipschitzian vector fields are given