Lipshitzian Vector Fields, Upper Gradients And Distributional Derivatives

We prove that given a locally integrable function f on an open set of an Euclidean space the distributional derivative $Xf$ with respect to a locally Lipshitzian vector field X is locally integrable if, and only if, the function f admits a locally integrable upper gradient along the vector field X; in this case $Xf$ coincides with the Lie derivative LX f and $|Xf|$ is the least upper gradient of the function f. Applications to systems of locally Lipshitzian vector fields are given.