Multipole distributions and hyper-fluxes

We outline here a simple mathematical introduction to the notions of multipoles for a general extensive property Π from the point of view of continuum mechanics. Classically, Π is the electric charge, but the theory is not limited to electrostatics. The proposed framework allows a simple computation of the bound “charges” and bound multipoles of lower orders. In addition, if the property Π has a potential function, a general expression for the mechanical force (power) functional acting on bodies containing the property is presented. Finally, using a similar viewpoint, we consider hyper-fluxes—flux fields of tensorial order greater than one—and show that moving multipoles (in particular, a moving dielectric) give rise to hyper-fluxes.