Electromagnetic quantities in 3-space and the dual Hodge operator

In 3-space, polar vectors (such as a force F) must be distinguished from axial vectors (such as a torque C). A more detailed analysis shows that ´axial vector´ C is an antisymmetric tensor of the second order. In 3-space, its 9 (= 3²) components C_ij are grouped together in a 3×3 square. In presentations of electromagnetism, taking account of the existence of these two types of vector, two options exist (α and β), vectors E and B being associated either with group D_αH_αj_α or with group D_βH_βj_β. Option α (developed in the book Global Geometry of Electromagnetic Systems by Baldomir and Hammond, Clarendon press, 1996) is based on the use of differential forms: magnitudes in m^-1 lead to the polar nature of E and H_α while magnitudes in m^-2 correspond to B, D_α and j_α. The * Hodge operator then becomes necessary allowing, in particular, the forms *E→E_H (the H suffix indicating the role of the Hodge operator), to be able to write D_α=ε₀E _H in the case of the vacuum. This operator is also vital for linking B to H_α, and j_α to E. In this paper, we defend option β, showing that the choice of D_β , H_β, j_β is justified by purely geometrical arguments; no special operator is then required to establish links between E and D_β, E and j_β, B and H_β