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II. Exterior Differential Foundations of Electrodynamics

Hehl and Obukhov start with the most natural first axiom namely that of conservation of electric charge. We write it here in the form

(1)

where

(2)

are the charge density 3-form and the current 2-form respectively. Using Stokes’ theorem, in equation (1) the following equation results

(3)

which is equivalent with the local form

(4)

in view of the arbitrariness of the volume V. Here the wedge symbol after differential operator shows that we have to do with an exterior differentiation. Now, using a known property of this operation, just because the charge density is a 3-form, we must logically have

(5)

so that there is a 2-form D, such that

(6)

Taking this equation into (4) gives

(7)

showing that the 2-form under exterior differentiation is actually the exterior differential of a certain 1-form H:

(8)

The results thus far can be summarized as a first set of Maxwell equations representing the Gauss and Oersted-Ampère laws:

(9)

where D is the electric excitation and H is the magnetic intensity.

This presentation makes it plain that a first set of Maxwell equations can be simply deduced as logical consequence from just a single experimental axiom – that of charge conservation – with  proper definitions of quantities of interest. These definitions are to be offered in the language of exterior differential forms. We strive to show here that using exterior differential forms has not only an importance of convenience, but also one of deep essence. Let us follow the presentation of Hehl and Obukhov. It goes on with the definition of the Lorentz force 1-form as the second axiom

(10)

where e is the particle charge, its velocity vector and F, E, B are the differential forms of force, electric strength and magnetic flux:

(11)

The definition of Lorentz force basically introduces these differential forms throughout their experimental correlation as stated by Lorentz. The symbol () represents here the inner product of a vector with a 2-form, the result of which is a 1-form. Mention should be made that the vector components must have the variance of the coordinates, i.e.

(12)

With this introduction of the field strengths, the third axiom states what our authors call the magnetic flux conservation

(13)

Applying here Stokes’ theorem again, we get the local form of the equation as

(14)

wherefrom, by exterior differentiation, we get the equation

(15)

expressing the fact that magnetic strength represents a field without source. The results can thus be summarized as a second set of Maxwell equations representing the Faraday induction and the closure of magnetic field strength laws:

(16)

From this moment on, by supplying adequate constitutive relations, the electromagnetism can be developed, as it really was, in any of its directions.

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