The description of the electromagnetic field with four vectors, E, D, B and H, has always puzzled me. Why four and not two? The explanation that the two real field are E and B with D and H being two auxiliary fields helping you taking into consideration the complications due to the presence of matter, seemed not to be the final word. I think of it from time to time; here is my actual position, my "phase space" about this subject.
1) First you need to introduce differential forms
How and why? It isn't an unnecessary complication?
You introduce differential forms from a logical point of view, as always in mathematics. You extend previous concepts fixing logical rules (for this I like the explanations of Richard Courant and Herbert Robbins in "What Is Mathematics? An Elementary Approach to Ideas and Methods").
The starting point is the concept of differential (I find clear the explanation in Demidovich). For the introduction of differential forms you can see for example "Calculus: a complete course" Adam and Essex, Chapter 17: "Differential Forms and Exterior Calculus". In brief you start considering differentials, and note that they have properties similar to vectors (you can see it for example considering the differential of the function f(x, y, z) and its gradient ∇f). Than, proceeding step by step, you build new objects (differential forms) and new operations (wedge product and Exterior derivative).
2) Than you understand that electromagnetic field is better described using differential forms
Here we face with one of the argument I love more: the relation beetween physic and mathematics. As an introduction I can't resist quoting Wigner:
"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (Eugene P. Wigner from The Unreasonable Effectiveness of Mathematics in the Natural Sciences,1960).
I think one can see a bit of this in action with the introduction of differential forms in electromagnetism. The best introduction I found is the one of Alain Bossavit (Electricite´ de France) in a series of articles on the geometry of electromagnetism (published in 1998 in J. Japan Soc. Appl. Electromagn. & Mech. 6) you can find on the net, in wich he focus on the fact that "one can discuss the same physics within widely different mathematical formalisms". He shows that different geometrical objects can serve in describing electromagnetism. In his words:
"The World is, and it certainly has order and structure. But order and structure in our descriptions of the world are something else, even if we try our best towards a close match, in the process of model building. Pure mathematicians try to discover, analyse, and classify all logically possible abstract structures. People who apply mathematics, including physicists and engineers, use them to construct specific abstract structures, which reproduce some of the features of the real world, and thus can help in explaining or predicting the behavior of some definite segment of reality. So mathematical entities by which we thus describe physics are not a priori frames of our thinking. They are our creation, moulded of course by the structures of the world out there, but still abstract things. Therefore, they are more or less adequate as tools with which to deal with the real world, which means one can—and one should—criticize the way they are applied, and question their adequacy".
He introduce the use of differential forms for describing electromagnetic fields and explain concepts I always found a bit obscure such as pseudo vectors, twisted forms and orientation of space in a very clear manner. (to be continued ...)
