  The topological properties of a material's electronic structure are encoded
in its Berry curvature, a quantity which is intimately related to the
transverse electrical conductivity. In transition metal dichalcogenides with
broken inversion symmetry, the nonzero Berry curvature results in a valley Hall
effect. In this paper we identify a previously unrecognized consequence of
Berry curvature in these materials: an electric field-induced change in the
electrons' charge density orientation. We use first principles calculations to
show that measurements of the electric field-induced change in the charge
density or local density of states in MoS$_2$ can be used to measure its
energy-dependent valley and orbital Hall conductivity.
