The derivation of Laplace’s and Poisson’s equations assumed constant permittivity, but there are cases of spatially varying permittivity in which the equations will still apply. Consider the vector identity, ∇ · (ψG) = G·∇ψ + ψ∇ ·G, where ψ and G are scalar and vector functions, respectively.

Determine a general rule on the allowed directions in which ∈ may vary with respect to the local electric field.