  We discuss the closability of directional derivative operators with respect
to a general Radon measure $\mu$ on $\mathbb{R}^d$; our main theorem completely
characterizes the vectorfields for which the corresponding operator is closable
from the space of Lipschitz functions ${\mathrm{Lip}}(\mathbb{R^d})$ to
$L^p(\mu)$, for $1\leq p\leq\infty$. We also consider certain classes of
multilinear differential operators. We then discuss the closability of the same
operators from $L^q(\mu)$ to $L^p(\mu)$; we give necessary conditions and
sufficient conditions for closability, but we do not have an exact
characterization. As a corollary we obtain that classical differential
operators such as gradient, divergence and jacobian determinant are closable
from $L^q(\mu)$ to $L^p(\mu)$ only if $\mu$ is absolutely continuous with
respect to the Lebesgue measure. Lastly, we rephrase our results on multilinear
operators in terms of metric currents.
