Sure, a field of a vectorial quantity can originate from a monopole. So are the electric fields and electric charges.

But it cannot be for a gyratorial quantity, such as the field %breve B.

The theorem of the hedgehogs is against it: it is impossible to entirely comb a haired sphere.

The magnetic potential %vec A is linked to the currents. No monopoles again.

Thank you, especially Emmanouil with your educative comments; and Jacques with the specifics. Mathematics only represents & models physical realities, which eventually when proved change our perspective & knowledge altogether.

Like pointed out in the article & above thread, & further have been very wonderfully & precisely explained by Emmanouil, we will have to take into account that we needed a proper technique with detection & measurements of magnetic vortex fields. The iron filings were not mapping them correctly, and the reasons have been well explained by scientists who developed ferrolenses/Ferrocell devices, along experimental profiles measured by Emmanouil and the whole project team that I was given honor to visit & see really at their Antonidakis Laboratory @TEI, GREECE. Further experiments they might conduct at CERN may finally prove beyond any skepticism about monopole. My Modified Maxwell Equations will standby to explain theoretically wholistic process, hand shaking with experimental findings proofs that will explain about singularities GUTs point to monopoles. Logically one will have to come with sources energies that are viably generated & monopoles provide that. Also the sinks are also provided by monopoles that will establish the energy flow that will create space-time energy matter over and over may be expanding to infinitum!!!!! We will await experimental confirmation with the monopole magnetic action to unify field theory.

Pierre Curie (1859-1906) : Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique et d'un champ magnétique. Journal de Physique, 3e série, t. III, 1894, p. 393.

And on the wiki:

http://deontologic.org/geom_syntax_gyr/index.php?title=Les_quart-de-tours_entre_vecteurs_:_gyreurs

Thank you especially Emmanouil with your detailed explanations, & Jacques with great reference please send a translation version of Pierre Curie's paper in English appreciate quite highly great its original work there.

Sincerely,

Rajan Iyer

In 1894 Pierre Curie did not know the existence of the right mathematical tool, established by Gregorio Ricci-Curbastro. He only had the shorthands used then in crystallography. Only in 1921, in his first conference in Princeton, Albert Einstein publicly wrote the nine coordinates of the antisymmetric tensor %breve B. It is simply the restriction in 3-D of the electromagnetic tensor in 4-D.

The paper of 1894 is reprinted in 1984 by Editions des Archives contemporaines, pages 118-141 of "Oeuvres de Pierre Curie".

Vae victis ! There is no translation either of the original papers of Erwin Schrödinger. One must read them in german. Carl Ramsauer also must be read in german.

A pupil of Bela Bartok was surprised that the master indicated him a danish musicologist, never translated from danish. Bartok replied, as laconic as usual: "With a dictionary, one can read anything".

When I studied @institution with German collaboration, I got the feeling that somethings aren't possible to be translated, since real essence won't get transferred per se in the process. Even if that's the reason here and/or if a reasonable interpretation acceptable by scientific community is possible, then it will help going to original precept, logic, & argument operating theory will get revealed helpfully.

In summer 1958, my father was struggling with Karl Marx and Friedrichs Engels, and the difficulties to translate and understand some german verbs used by Hegel for dialectics. And he was a poor diver, so never could dive near two breams, who scoffed at him at nearly eight meters deep. So one night, he dreamed of the two breams, with the beards of Marx and Engels.

Of course German is one of the toughest languages to translate. We can only hope that our thought logic might match at a page to unravel unified knowledge.

Modification of enm theory including the magnetic monopole (yet to be found)

is alredy discussed in Jacksons book. Even if included, this would not alter conservation of energy. Then so far div(B)=0, no sources or sinks for B field.

The operator "div" applied to an antisymmetric tensor (like B, %breve B) does not give the result you expected.

div means divergence and most commonly applied to vectors. The result

of the operation is a scalar. When the result is zero you have no source or sinks.

B here is a vector field, (div (B)=0 is one of the Maxwell equtions) and not the antisymmetric field tensor of enm

To understand source sink mechanism with monopoles, a CERN experiment with Higgs Boson Einstein condensate may bring out, if synthesizable tuned to that, monopoles skyrmions that isolate these might systematically measure the magnetics coherence with electrics. By separating monopoles through special techniques such that magnetic effect and the flow associated can be studied, that then will tell whether cancellation of north & south poles occur & otherwise source sink mechanism continues.

Since these are observations recent, we can't say the divergence applied to antisymmetric tensor will not occur with decoherent or incoherent magnetics with electrics. Scalar will indicate that it is measurable electromagnetic waves photons phonons real space energy matter.

The dahu is a magical and wild animal living in mountainous slopes. So his right legs are shorter than the left legs. To hunt the dahu, you come silently behind him, and you shout "Bouh !". Affraid, the dahu turns on himself, and falls.

For 1888 (Oliver Heaviside), the "vectorial product" or "cross product", and the misrepresentation of magnetic field B by a vector share the main property of the dahu : if you apply a symmetry ont it, it falls.

So the symmetries have been forbidden to the physicists. And the most tolerant-to-absurdity students were selected.

There are at least 3 different equivalent ways to express the Maxwell equations.

You are aluding to the second way, but its all the same. Mine, the best known, is 4 separte equations. The magnetic field is still a vector field, your way includes both electric and magnetic field in just 2 separate equations.

None of this means that the B field, the magnetic induction field, is nonsensical.

A deafs dialog...

Since Faraday, the group think teaches that the field B is of vectorial nature.

However, the group think remains wrong. James Clerk Maxwell was aware of that and noticed it in the § 15 of his Treatise. The proof by Pierre Curie is definitive, in 1894.

B remains a gyror, or antisymmetric tensor.

But the magnetic potential A is a true vector. In a plane-polarized electromagnetic wave, P, A, E, and B are coplanar. Not in a circularly or elliptically E.M. wave.

I fail to see why B is not a vector, since A is and B=curl(A), that is why

div(B)=0 automatically. The antisymmetric tensor is called F (mu,nu), contining both E and B vectors.

In an electromagnetic wave that you describe E and B are not coplanar, but at right angles to each other. One hopes that since the days of Maxwell, the notation is consolidated and mistakes have been corrected.

ALL modern sources agree here. Only some use MKS(coulombs) and others cgs or Gaussian units.(electrostatic units of charge). This changes in minor detail what some formulas look like.

Maybe you term B a gyror because of the Lorenz force law, a force perpendicular to both velocity v and magnetic field induction vector B.

This does not convert B into a tensor.

Perhaps inside some material medium you may have to use tensors, in some situations where electromagnetic properties vary as direction in a crystal lattice.

div(B)=0 would be modified to have a surce (finite right hand side) in the case of magnetic monopoles, similar to div(D)=rho

B=curl(A) would also have to be modified.

Already in 1921, first conference in Princeton, Albert Einstein wrote the restriction B in 3D of the 4D electromagnetic tensor. With its nine coordinates.

Already in 1894 Pierre Curie had showed that B has none of the symmetries of a vector. It always has the opposite symmetries.

I have already given the complete course for teachers:

http://deontologic.org/geom_syntax_gyr/index.php?title=Les_quart-de-tours_entre_vecteurs_:_gyreurs

The curl of a vector field is never a vector. It is always a gyror.

But the div of a gyror field is a vector.

Juan,

"I fail to see why B is not a vector, since A is and B=curl(A), that is why ..."

As far as I know B is considered by the literature as a pseudovector.

But alas most people understand "pseudo" as it was "quasi", and not "false".

Thank you all with your great comments with specifics, especially Professor Antonidakis & accompanying discussions. These will get revisited more & more.