Chapter One

Sooner or later, any discussion of basic electromagnetic theory is certain to come to the issue of how best to categorize the vectors B, the magnetic induction, and H, the magnetic field strength. Polar or axial is the central issue. From an elementary physical perspective, taking B as an axial vector seems appropriate since, as far as we know, all magnetism originates from currents (see Appendix 14.2). From a mathematical standpoint, however, taking H as a polar (or true) vector seems a better fit with an integral equation such as Ampere's law, ∫H · dl = μ0I, particularly in relation to the subject of differential forms. But taking the view that B can be one sort of vector while H is another seems to be at odds with an equation such as B = μH in which the equality implies that they should be of the same character. A separate formal operator is required in order to get around this problem, for example, by writing B = μ * H where * converts a true vector to an axial one and vice versa, but for most people, any need for this is generally ignored.

Geometric algebra provides a means of avoiding such ambiguities by allowing the existence of entities that go beyond vectors and scalars. In 3D, the additional entities include the bivector and the pseudoscalar. Here the magnetic field is represented by a bivector, which cannot be confused with a vector because it is quite a different kind of entity. Multiplication with a pseudoscalar, however, conveniently turns the one into the other. But the new entities are far from arbitrary constructs that have simply been chosen for this purpose, they are in fact gene