Orthogonal mappings of fields and sources

Abstract

In previous chapters we considered the mapping of a region into itself if the medium possessed a plane of reflection or conjugation symmetry. In the one case the mapping yielded an equivalence relation, in the other a reciprocity relation. In this chapter we shall generalize the previous procedures in two essential ways. First, we shall no longer confine ourselves to media possessing a spatial symmetry structure, but shall compare two different spatial regions in which the constitutive tensor in the one can be mapped from the given, or from the Lorentz-adjoint constitutive tensor in the other, by means of an orthogonal spatial mapping (rotation, reflection or inversion). Second, the orthogonal spatial mappings will no longer be restricted to reflections or to rotations through an angle π, but will be extended to include the ‘full rotation group’ [61, Sec. 2–7] which comprises rotation through arbitrary angles, with or without reflection or inversion. Conceptually there will be little new in these generalizations, but they will permit the formal systematization of the ideas developed till now.