Rinat,

You said that Maxwell derived electromagnetic attributes from a bulky and ridiculous model of ether vortices.

It is not so. He derived nothing except integrate the electric and magnetic attribute previously discovered by Ampere, Faraday, Gauss and Coulomb through experimentation, as I mentioned before and as can be verified by anyone from undergrad textbooks that explain how these experimentalists from the past discovered electric and magnetic behavior, such as "Physics" by Halliday and Resnick or "University Physics" by Sears Zemansky and Young.

He linked the velocity of light to eps_0 and mu_o as explained, not by some chance miracle.

Maxwell's attempt as trying to match known electric and magnetic attributes with some ether was simply trying to match them with the theories of the time about the possible nature of space. We know better today. He did the best he could at the time.

  Dear Mohammad,

  E and B are components of the electromagnetic field. In Maxwell's equations they appear as the variation in space of one gives the variation of time of the other (Faraday and Ampere's laws). In both cases their interaction or coupling is with electric charge only by means of the photons. Meanwhile the weak interaction is a short range interaction  where the fields are coupled to their sources (quarks) by means of the W+.W- and Z0 bosons, which are very massive particles in opposite of what happens with the photon whose rest mass is zero. The equivalent electric charge associated to U(1), is the flavor for the weak interaction with inner symmetry SU(2). Thus they are absolutely different interactions, while we touch everyday photons and their interactions in chemistry or biology,  the weak interaction is very  almost restricted to nuclear reactions.

  In the Standard Model it is possible to see that there is a spontaneous braking of symmetry of which unified both interactions at a high energy. Thus the things are absolutely different of what happens with the electric and magnetic field in Maxwell’s or Lorentz’s equations for the electromagnetic interaction which only have the coupling electric charge-phonon.

I hope that this can helps.

Sir, I feel honored of your response, and also it helps to clear my understanding… Still, don’t you think that your clarification goes against the spirit of grand unification (the Physicists’ Grand Endeavor)?

N.B. I do understand that physicists are trying to unify gravitation and strong force with electroweak interaction and I am also aware that my question is a backtrack (!).

Why don't just remember, that E & B are parts of F(mu,nu) tensor? And generally you can't divide it into E & B. In fact this division works for electrostatic and magneto-static case because of structure of Maxwell equations: one zero eigenvalue.

Dear Mohammad,

I am fully agree to try to unify the four interactions that we know nowadays. Electromagnetism is in fact unified with the weak and almost with strong. Which is out of this unification, for the moment, is gravitation.

In principle the unification is quite logic if we think that everything has come from only one point in a given "spacetime".

But unification of "interactions" is a very different thing that to consider the parts (components) of the electromagnetic field as a part of this unification. If you want to generalize your concept of the fields, you can go to the Yang-Mills equations which have electric and magnetic non abelian fields, and this is not also any kind of unification but a form to write in compact form the fields provided by certain symmetries, U(1) for electrodynamics, and SU(N) for the Yang-Mills equations.

The gauge interpretation of these equations introducting concepts as the spontaneous breaking of symmetry is what allows to see how these interactions are glued. Unfortunatally Einstein's equations of gravitation are not into this context instead they are gauge fields too.

Four Maxwell's equations are

These equations summarize all of classical electrodynamics. Furthermore these equations can be written in a compact form in terms of, the tensor,

Fμν=∂μAν-∂νAμ, where, Ei=c F0i and  Bi=εijk Fjk

This is the first successful attempt to unify natural laws, in this case the are laws of electrodynamics. Quantum version this theory was developed by Feynman, Tomonaga and Schwinger, for which they received Nobel prize in 1965

In my understanding, according to Dr. Eugene Prokhorenko, other than the static case there are no different entities as E and B. According to Dr. Daniel Baldomir, the covariant formulation of classical electromagnetism is only a compactification rather than unification. And to Dr. Biswajoy Brahmachari, the solution rooted in the unification of classical and quantum physics (!). I would like to thank you all for your responses.

Actually, E(x) and B(x) are vector fields having one index, which are determined via four Maxwell's equations at the classical level. Now they can be combined in the field tensor having two indices. If you want to know exactly how E(x) and B(x) are embedded in the Fμν(x) tensor see the book by J. D. Jackson. It is (in some sense) analogous to two columns forming a matrix, but not exactly so.

Now the two equations given above describe how to determine the field tensor. Therefore four Maxwell's equations are equivalent to two equations satisfied by field strength tensor Fμν(x) . This is an unification (or perhaps I can call it "synthesis" as commented by Sasso below) of the laws of electricity and magnetism at the very classical level in the sense that now number of equations have reduced from four to two. This description is also reversible, that is you can always rewrite them in component form which will give back Maxwell's equations.

Next question is how formulate this theory in such a way that it is also valid at the atomic and subatomic scales. This is done by the formulation of Quantum Electrodynamics (QED). I hope I could explain my point.

Vector fields E and B with respective forces represent different physical realities also if they are correlated. The unification of these two vector fields is impossible because they represent different phenomena. What we can say is that a physical symmetry of analogy exists relative to the dynamic behaviour of the two fields because the variation of E causes a field B and a variation of B causes a field E. Maxwell' s equations don't represent an unification of fields E and B but they represent only a synthesis, not alwais rational, of laws of electromagnetism.

I would look at this question as follows: first you have a Lorentz tensor, F^{mu, nu}. It is one single object, and there is no more reason to make a difference between the 01 component and the 23 component than there sould be to distinguish between, say, the first and the third component of a vector. The distinction between magnetic and electric field is wholly arbitrary.

But wait, someone might say, there often *is* a good reason to distinguish between the first and third component of a vector. If I have to go from A to B, it is not the same whether the coordinates of B with respect to A are (0, 0, 100) meters, or (100, 0, 0): there is a real difference between going uphill and horizontally.

Just as gravity introduces a difference between the coordinates of a vector, so does time in a Lorentz tensor: most Lorentz transforms in real life couple time and space only weakly. So it actually makes (partial) sense to limit oneself to Lorentz transforms that do not couple space and time at all. Such are, of course, simple rotations.

If we now look at the F^{mu, nu} tensor as it transforms under rotations, we see that it decomposes in two independent parts: the F^{0 k } and the F^{k l}, where k now goes only from 1 to 3. The two terms are independent, and both transform as vectors: the first one as a real vector, the second as a pseudovector.

so as long as we only consider rotations, electric and magnetic fields are altogether separate. But Maxwell's equations are in fact invariant under a larger symmetry group, namely Lorentz'. If we transform under arbitrary Lorentz transforms, nothing remains of the difference between electric and magnetic field.

This way to consider physics is an old and obsolete way that luckily nobody will consider in future.

@ Daniele: I look forward to your version of non-obsolete physics. I feel some obligation to describe things in a way which I believe can be defended as being true, if perhaps old-fashioned.

@ Dr. Daniele Sasso ~

Sir, weak and electromagnetic interactions were also two different realities, but they have been unified in an ad hoc fashion (Higg’s mechanism)... what is your say? I am getting puzzled with the words 'compactification’ and ‘unification’.

Equations that historically were defined "Maxwell's equations" are equations that were written in integral shape in B. Brahmachari' s comment (pag.1 of the question).

I understand also B.B's consideration about the possibility to reduce equations from four to two, but I am opposed in general to mathematical superstructures that hide the physical meaning of equtions that is instead clearest in the historical formulation, even if I prefer the differential form to the integral form. Moreover it needs to specify I don't subscribe fully the historical formulation of equations because, for instance, the equation 2 isn't in actuality an equation but it is rather an identity. In fact in the group of " Maxwell's rationalized equations" I replaced that equation with Lorentz's force that seems to me very important in order to complete the group of relations that are useful for describing also interrelations between electromagnetic magnitudes and mechanical magnitudes. It is manifest that I am criticizing the present trend to build mathematical superstructures because it implies in general clouding of the physical meaning of phenomena. Similarly I think the concept of vector field is clearest from the mathematical and physical viewpoint while the concept of scalar tensor is for me a mathematical superstructure that hides the physical meaning and isn't necessary. We are talking here about physical problems, the situation would be certainly different if we could talk about mathematical problems.

With regard to question that is raised by M.S. Mahmood about the unification and compactation, I think the concept of compactation is closest to the concept of synthesis while the concept of unification implies a reduction of two or more concepts to one.

I repeat with big security and objectivity that it isn' t possible to unify concepts of electric field E and of magnetic induction B because inter alias E is measured in V/m and B is measured in Tesla. On the question of the unification of weak and electromagnetic interactions there is much to say, but another question would be opened that regards the Standard Model and the Non-Standard Model.

Hence, can we say that E and B are two different phenomena and only be unified in the empirical sense by introducing motion?

Please, take care with the definitions, over all with a subject as electrodynamics which is very well established: classically and quantically.

The electromagnetic field is a physical entity, where the waves are formed by electric and magnetic field carrying energy and momentum. The same happens for particles as the photon.

The divergence of B equal zero is one experimental fact because there are no isolated magnetic poles and less is impossible to substitute it by the Lorentz motion equation which is not contained in the Maxwell equations. This equation carry other different physical information.

Mohammed, I wrote E and B cannot be unified into only one concept because they are two different physical magnitudes that are measured also with different units of measurement. What is certain is that E and B are correlated through an interesting physico-mathematical symmetry that is able to describe very well electromagnetic field and waves.

Daniel, I take care with definitions and also with other exigences of clearness. I agree with you, the e.m. field is defined by both electric field E and magnetic field B that are no doubt two different physical magnitudes and are correlated just through M.'s equations. The divergence of B is always zero for any value of B and for this reason I wrote it is an identity and therefore it is not necessary to introduce it into the group of M.'S equations.I proposed then to insert Lorentz's equation because that equation opens all a series of new problems relative to the interrelation between electromagnetic magnitudes and mechanical magnitudes. I did it in numerous papers of mine and for me that solution is good.

Christian, I agree, nature is one but it is varied. It is impossible to think only one equation or a limited group of equations is able to describe all the variety that is present in the nature.

Regards.

On the same grounds, it would be perhaps sensible to view E_x as being altogether different from and incomparable with E_y and E_z, the rotational symmetry which seems to connect them being merely one of these puzzling artefacts which mathematicians like to confuse us with. 

E_x is connected to E_y by rotation in the same way as the E vector is connected to the B pseudovector by Lorentz transform.

The lamentable argument concerning units shows a further distressing lack of familiarity with even that most elementary aspect of electromagnetism: reducing to basic units (something which should *always* be done if you want to argue about units)  electric fields are measured in

A^{-1} kg m s^{-3}

whereas the B field is measured in the SI using

A^{-1} kg s^{-2}

So an electric field is the same as a magnetic field multiplied by a velocity. To obtain the desired unification, you introduce E/c in one part of the tensor, and B in the other, or viceversa.No objections from units.

That Nature is one, and yet varied, is a sentiment I can agree with. But it is, of course, a physicist's task to display the unity where it is found. Unifying magnetic and electric phenomena via the concept of Lorentz invariance and electromagnetic tensor was an extraordinary first step in unifying the variety present in Nature.

Leyvraz, let you calm, you start with the fifth gear but like this your engine burns out. Let you persist in mistakes and you have also the impudence to define "The lamentable argument concerning units" demonstrating you don't know the question of units of measurement is the first issue in an elementary course of physics. Proofs of your mistakes are just in your words: "So an electric field is the same as a magnetic field multiplied by a velocity". Just that claim derives from Lorentz's equation and proves definitively electric field and magnetic induction field are two different physical quantities. The question for me is closed.

`` The question for me is closed.'' I agree.

c=1

a=3

b=a-c=2   ok

End of modern and postmodern physics.

Beginning of contemporary physics.

Mohammad

I always felt that unifying the E and B fields into a single nondescript electromagnetic tensor was not conducive to deeper  understanding.

Such unification in a single tensor constantly hides the fact that "E and B are components of the electromagnetic field. In Maxwell's equations they appear as the variation in space of one gives the variation of time of the other (Faraday and Ampere's laws). " to quote Daniel.

All the more so since electric fields are know to be related to the Coulomb inverse square interaction while magnetic fields are know to be related to the inverse cube interaction, as experimentally demonstrated as recently as 2014 in the first paper I quote.

Irreconcilable characteristics, that add to the fact that both fields induce each other, in Maxwell, which are details that constantly go unnoticed when "thinking" only of the "electromagnetic tensor".

If you are curious as to what sort perspective can be had from viewing them separately, I refer you to the second recently published paper I quote.

Other views are of course possible, but I think that "thinking" of both fields as being separate can be more productive than merging them in a nondescript concept such as the electromagnetic tensor. Find for general views, but no that much for exploring.

http://www.nature.com/articles/nature13403.epdf?referrer_access_token=yoC6RXrPyxwvQviChYrG0tRgN0jAjWel9jnR3ZoTv0PdPJ4geER1fKVR1YXH8GThqECstdb6e48mZm0qQo2OMX_XYURkzBSUZCrxM8VipvnG8FofxB39P4lc-1UIKEO1

http://www.omicsonline.com/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.php?aid=70373

I agree that in a low-velocity "Galilean" approximation, there is something to be said for viewing the two fields separately. That is in particular the case for electro- and magnetostatics, to which you clearly refer. But whenever radiation becomes important, it is a trifle distressing to have two altogether unrelated fields propagating orthogonally one to the other, both having equal magnitude (in spite of the different units, Daniele!). Similarly, when such an electromagnetic field acts on an electron, particularly on a relativistic one, it is  not terribly helpful to separate the field in electric and magnetic.

In other words, at a *very* introductory stage, when all time dependence is essentially neglected, it may not be wholly meaningless to make such a separation. For any significant understanding of the more difficult aspects of electrodynamics, though, such a view is altogether a hindrance.

Agreed that this is absolute first stage.

Clearly the triple orthogonal relation being maintained during propagation is unsettling at first. Easier to deal with if seen as conceptually applying to isolated quanta remaining permanently localized.

However, pushing one step further with the separate fields approach, it can be shown that it possibly allows an explanation of all electron velocities, including asymptically close to c. So at the elementary particle level, the whole range of velocities can apparently be covered:

http://ijerd.com/paper/vol6-issue4/A06040110.swf

to Dr. André Michaud ~

I feel that unification is necessary to get an operational hand to serve technological purpose. And compactification is useful for theoretical anatomy (well developed for EM). We have readymade technological solutions in the case of electromagnetism, so this could be a good specimen for tailoring theoretical unification (merge different concepts) to peer inside natural rules. Simply this was my motive.

A fundamental feature of electromagnetism which is captured in the tensor representation is the principle of duality between electric and magnetic quantities. The concept of duality lends itself to various generalizations under the names iso-duality and geno-duality which have been discussed in such articles as my Jan 2013  "Geno-Bragg's law and 10*x10 representation of SU(3) symmetry for quasicrystal structures in Journal of Computational Methods in Sciences and Engineering.

Electric and magnetic fields are simply two cases of the more fundamental electromagnetic field.  Both are required to exist by special relativity.  Simply perform a Lorentz transformation on the Coulomb force and you get the Lorentz force.

It is easily demonstrated that the Lorentz force is identical to the Coulomb force (inverse square interacton). The true inverse cube magnetic interaction force is not covered by any of the current force equations:

http://www.ijerd.com/paper/vol6-issue6/F06062734.pdf

Dear Mohammad,

Let me to try to summarize some points that, I think, there were a little bit confused in the discussion:

1.       Maxwell only introduced the displacement current in the old Ampere law of magnetism. This was what allowed to couple the electric and magnetic equations in a simple wave equation, whose velocity depends only of the permittivity and permeability in vacuum. The fantastic thing was that this velocity coincides with the one of light. Thus in fact electromagnetism of Maxwell could be said that had made one unification of subjects such as Electricity-Magnetism-Optics in vacuum.

2.       The motion of the light in Optics was full of problems. For instance Fizeau detected a  dragging effect when the light propagates in medium in motion respect to the "aether", but the magnitude of the effect that he observed was far lower than expected. The problem was that he assumed the galilean transformations instead of Lorentz ones, where electricity and magnetism were absolutely indendent quantities. In fact he added the velocities directly as a sum of simple vectors.

3.       The problem reached its highest difficulty to interpret when Michelson and Morley tried to calculate the velocity of the light using the Earth motion repect to the "aether". Their interferometer said that the addition of velocities where not right if the light where involved in the motion.

4.       Einstein published a seminal paper on the electromagnetic motion of the fields using the transformations that Lorentz and Poincare have had found some time before. In those transformations, the electric and magnetic field are just components of a Lorentz second rank tensor. This means, e.g, that if you have one electric charge at rest, then you only see electric field, but other observer in relative motion will see magnetic field too besides the electric one.

5.       This means that the relation of electricity and magnetism is not a theory, but a real behaviour of the electromagnetic fields. What is important, and this is the main achievement of Einstein, from my humble point of view, is that this is not only a behaviour of electromagnetism but of the rest of physics too. That is to say, the Special Relativity theory.

6.       Thus electric and magnetic fields are obviously different physical objects, but components of an electromagnetic field Lorentz tensor on the space and time geometry. Notice space and time or space-time have not usual Euclidean metric and therefore the components play a very different role than the usual vectors.

One simple achievement is that there are two field invariants which share every observer

            B2 -   E2/c2 =Lorentz scalar

            E.B= Lorentz pseudo-scalar

7.       This unification in Physics is very far of what is understood nowadays by unification of interactions, as electroweak introducing the unitary symmetries SU(2)xU(1), and so with the new color or flavor charges.

       I hope that this can help     

Thank you Dr. Daniel and thanks to all others for your answers.

 One can just add that in 1861 James Maxwell tried to unify electricity and magnetism in the most apparent sense deriving their atributes from a bulky and ridiculous model of ether vortices.He also tried to calculate the speed of light using some dubious suppositions and making many mistakes, as P.Duhem and D.Siegel later disclosed.It was a miracle that from his cumbersome model Maxwell managed to arrive at c=300000

Rinat

Maxwell did not derive electromagnetic attributes from a bulky and ridiculous model of ether vortices.

His first equation is Gauss' law for electricity. His second equation is derived from Faraday's law. His third from Gauss' law on magnetism and his forth is a generalization of Ampere's law. What he did was unify into one coherent integrated theory all these experimentally confirmed laws that were not clearly linked to each other previously. The idea that electromagnetic energy propagated as waves involves by structure that a medium had to be involved. This is why he used the idea of ether.

Moreover, it was no miracle at all that he could calculate the actual speed of light of exactly 299792.458 m/s (not 300000). He simply was very insightful and competent with mathematics.

The manner in which he succeeded in linking the speed of light to the permittivity and permeability constants  of vacuum is explained here:

http://ijerd.com/paper/vol7-issue4/G0704032039.pdf

Andre, may I advice You to read Daniel M.Siegel's masterpiece "Innovation in Maxwell's electromagnetic theory: molecular vortices, displacement current, and light" (Cambridge University Press, 1991). According to Siegel, the basic approximation that Maxwell made in his velocity-of-light calculations involved approximating the shapes of the vortex cells to spheres in some parts of the calculations, while elsewhere making assumptions that required the shapes of the cells to deviate from perfect sphericity.Moreover, beyond these approximations of which he was aware, Maxwell made one clear error in his calculations, as was pointed out by Pierre Duhem in his book on Maxwell's theory.Given this situation, one would have to consider it a great stroke of luck for maxwell to have come up with an answer for the velocity of wave propagation that was within 1 percent of any given number.

Besides that, the following words of Maxwell on his vortices model and on the electric particles that constituted an important part of it are rather blatant:  “these particles, in our theory, play the part of electricity. Their motion of translation constitute an electric current, their rotation serves to transmit the motion of the vortices from one part of the field to another, and the tangential pressures thus called into play constitute electromotive force. The conception of a particle having its motion connected with that of a vortex by perfect rolling contact may appear somewhat awkward. I do not bring it forward as a mode of connexion existing in nature” (Maxwell [1861-1862/1890], 1952, 345).       

Rinat,

You said that Maxwell derived electromagnetic attributes from a bulky and ridiculous model of ether vortices.

It is not so. He derived nothing except integrate the electric and magnetic attribute previously discovered by Ampere, Faraday, Gauss and Coulomb through experimentation, as I mentioned before and as can be verified by anyone from undergrad textbooks that explain how these experimentalists from the past discovered electric and magnetic behavior, such as "Physics" by Halliday and Resnick or "University Physics" by Sears Zemansky and Young.

He linked the velocity of light to eps_0 and mu_o as explained, not by some chance miracle.

Maxwell's attempt as trying to match known electric and magnetic attributes with some ether was simply trying to match them with the theories of the time about the possible nature of space. We know better today. He did the best he could at the time.

Obviously Maxwell had no idea of what the electric charges were electrons and positive ions in most of the materials. He also didn't know vectorial calculus, and he thought that the matter was a kind of fluid. From todays point of view the physics is quite difficult to follow, but what is fantastic is that he was absolutly right in the electromagnetic behaviour.

One of the most beautiful parts, from my humble point of view, is chapter XX of the second volume (electromagnetic theory of light) of his treatise on electricity and magnetism, titled ELECTROMAGNETIC THEORY OF LIGHT. He not only gives the good expression of the light but his value taken by the experimental results of Fizeau, Foucault, Weber or Thomson. All of them with different values.

Reading the two volumes is hard task but one can understand immediately that he knows all the electromagnetism of his time in a very deeply form. I do not think that we can say that he was wrong, he was ignorant of the atomic structure of the materials.

Daniel,

I agree with you.

I think he did the best job that could possibly be done with what had been understood of electricity and magnetism from the macroscopic level experiments that could be carried out in his time.

Any hypothesis that could be had at the time about the submicroscopic level could only be total guesswork, such as his hypothesis about ether.

A pure mathematical alternative exists to Maxwell equations. That alternative is formed by a complete set of quaternionic partial differential equations. It includes first and second order differential equations. It can handle point-like artifacts that disturb the continuity of the basic fields. The set shows that the set of Maxwell equations is incomplete. Maxwell equations apply coordinate time, where the quaternionic equations apply proper time. In quaternion space coordinate time plays the role of quaternionic distance. Proper time is played by the real parts of the quaternionic parameters.

The quaternionic equations hold for any basic quaternionic field.  Basic fields may differ by the kind of artifacts that disturb their continuity.

http://vixra.org/abs/1505.0149

 Dr. Leunen, you echoed Dr. Baumgarten and explained the quaternionic framework in more physical sense… thank you for your easy clarification.

>...experimentalists from the past discovered electric and magnetic behavior...

Panagiotis

Thanks for the reminder. I had forgotten about his contribution. From the description of the mind set of the Greek scientific community in his time, we can see that if this mindset was generalized today, this would be hugely beneficial in every domain of scientific achievement:  

https://en.wikipedia.org/wiki/Thales

    If you go in the direction of  the greek culture, perhaps it is more related with this question their "material monism"

   https://en.wikipedia.org/wiki/Material_monism

Dear Andre, thank you.

Dear Daniel, thank you.

According to Plato [ Timaeus]  elemental traces [i.e. matter] existed in disorder before their harmonization.took place.

@ Dr. Charles Francis

Sir, could you please explain a bit more?

The quaternionic partial differential differential elucidate that the E, B, A and φ fields are in fact parts of the differential of a more basic field. At least two basic fields exist. They differ in the way point-like artifacts disturb their continuity. Thus. the homogeneous second order partial differential equations that describe these basic fields are similar.

http://vixra.org/abs/1603.0021

http://vixra.org/abs/1605.0268

http://www.e-physics.eu/MechanismsThatKeepRealityCoherent.pdf

The electromagnetic field tensor is antisymmetric which implies that one could unify it with a symmetric tensor representing the gravitational field (as envisaged in Einstein's grand unification) in the framework of lattice gauge theory, as in the attached file published in the Hadronic Journal in 2013.

Charles,

Why the flux law is an "electrical law"?

Charles,

on your web page

http://rqgravity.co.uk/wikka.php?wakka=QEDSub#Maxwells_Equations

you write, "Thus, in a treatment based on the magnetic laws are mathematical identities, not physical laws."

Some of the differential field equations are common to all fields, but Maxwell used coordinate time where other field theories use the equivalent of proper time. Einstein followed Maxwell's approach. Maxwell, Einstein and their adherents introduced the spacetime structure that has a Minkowski signature. Other choices use a Euclidean structure. Mathematics indicates that the Euclidean structure is a smarter choice. It brings compacter and therefore simpler equations.

The electric field and the magnetic fields are parts of a more basic field. More basic fields exist. They possess (are governed by) the same homogeneous second order partial differential equations.

It is possible to see fields as continuums that are eigenspaces of operators that reside in a non-separable Hilbert space. It is also possible to describe these fields by multidimensional functions. 

Maxwell equations describe only the behavior of a small part of these fields. In Maxwell's time only that part was discovered via physical experiments. Thus before deliberating about Maxwell fields it is sensible to learn more about general multidimensional differential calculus, about number systems and about Hilbert spaces.

     When you define the potential A you are introducing the magnetic equalities, therefore the physics of magnetism, obviously is contained in Maxwell's equations as physical equations.

     Another thing is that the electromagnetic field can be interpreted as a curvature where the potentials are the connections and the magnetic equations represent the Bianchi's identities of the curvature. But this is just another form of mathematically represent electrodynamics.