A class of vector identities relevant to the representation of the electric current density

A rigorous mathematical proof is given of a class of vector identities that provide a way to separate an arbitrary vector field (over a linear space) into the sum of a radial (i.e., pointing toward the radial unit vector) vector field, minus the divergence of a tensor plus the curl of an axial vector. Such a separation is applied to the representation of electric current densities yielding a specific form of the effective polarization and magnetization fields which is also discussed in some details.