Symmetry Arguments in Electrodynamics

Symmetry arguments are among the most useful methods in physics. In electrodynamics, for example, it is often possible to gain information about electric and magnetic fields from the symmetries of their corresponding charge/current densities. In order to place these principles of symmetry on a mathematically rigorous foundation, however, it is necessary to demonstrate that for a transformation of interest T, and a vector field F⃗, the following equalities hold: ∇ · (TF⃗) = T(∇ · F⃗) and ∇ × (TF⃗) = ± T(∇ × F⃗). In other words, it should not matter whether one transforms a vector field and then calculates divergence/curl, or calculates divergence/curl and then transforms the result. Though this concept is quite intuitive physically, its proof is rather involved. My research so far has been focused towards investigating the transformations and coordinate systems for which the above two relationships can be proved. To date, I have successfully proved the first equality for some limited situations in cartesian, cylindrical, and spherical coordinates. The goal of my research moving forward will be to develop proofs which allow as broad an application of these relationships (and thus the symmetry principles relying on them) as possible.