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I am trying to find a book on electromagnetism for mathematicians (so it has to be rigorous). Preferably a book that extensively uses Stokes' theorem for Maxwell's equations (unlike other books that on point source charge, they take Stokes' theorem on $B\setminus\{0\}$ with $B$ being closed ball of radius 1, but this does not work, as Stokes' theorem only works for things in compact support). Preferably if it mentions Dirac delta function, hopefully it explains it as a distribution (or a measure...)

P.S. This question is posted because there are no questions about electromagnetism books for mathematicians. I have background in mathematics at the level of John Lee Smooth Manifolds.

A rural reader
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chhan92
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  • In addition, I have seen in Special Relativity have a 2-form called action and saw that electric field and magnetic field can be read from by looking at specific coefficients of its components. However, I do not understand how and why you can read these fields from that in that specific coordinates. To specify, I meant from here: http://www.math.toronto.edu/~drorbn/classes/0708/GeomAndTop/Maxwell.pdf – chhan92 Jan 12 '13 at 03:14
  • @Christopher White please read my above comment – chhan92 Jan 12 '13 at 03:35
  • http://www.thp.uni-koeln.de/alexal/pdf/electrodynamics.pdf – Isomorphic Sep 04 '14 at 16:44
  • The link $\uparrow$ given by @Isomorphic is dead now, but the Wayback Machine has a copy archived here. A simple Google search also yields what appears to be an updated version. – Urb Dec 21 '20 at 18:32
  • There was a discussion about including/excluding surfaces in volume integrals see https://www.physicsforums.com/threads/do-volume-integrals-involve-bounding-surfaces.1016139/ – Cheng Dec 05 '22 at 11:28
  • Mathematical book on "Maxwell" equation – Cheng Dec 06 '22 at 23:33

3 Answers3

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Scheck's books are mathematically much more precise than the average physicist's textbook.

Urb
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Franz Hochinger
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  • I have taken look at that book and it does not seem to prove Maxwell's equation from Biot-Savart's law and etc (as how Maxwell would approach from historical point of view?) – chhan92 Jan 12 '13 at 03:13
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Edit: as I re-read your question, it sounds like that's not what you're looking for: you want classical vector-calculus-based E&M, done right. Not sure how to help you there, although I still heartily recommend Misner, Thorne and Wheeler in general.

You might try chapters three and four of Misner, Thorne, and Wheeler's Gravitation, if you can find it in a library. (You'll want one and two as well, for background.) In those chapters they develop the basics of electromagnetism from the point of view of differential forms.

They do not attempt to be rigorous, but (as far as I can tell) that's a matter of choice, not ability: I get the sense that they thoroughly understand the niceties of the math behind what they're doing, but (since they are writing for physicists to whom that's not terribly relevant) they don't present it.

Urb
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Christopher White
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  • Thank you for good comments. It is just I was looking up Maxwell's equation in Griffth&Harris and later in wikipedia I figured out that dirac delta is indeed a distribution (this fact was nontrivial, as my interest is more onto differential topology and abstract algebra (not sure on analysis)). Before I move further, I was just hoping for good EM book done right, maybe giving full rigorous proof of important results that I might possibly get stuck in future. – chhan92 Nov 24 '12 at 06:57
  • Actually best thing is EM in more general setting if that is possible (maybe what happens to Maxwell's equation if our ambient space is any smooth manifold, instead of usual Euclidean space?). EM on low-dimensional manifolds would be cool, but if it does right on Standard Euclidean space, that is fine with me. – chhan92 Nov 24 '12 at 06:59
  • This may be totally naive, but if you're working on an arbitrary smooth manifold, then shouldn't ME transfer to equations on the manifold via the coordinate charts (at least locally). If the manifold is compact, then you can probably extend globally. –  Dec 04 '12 at 07:46
  • William: I think you're right, but the thing is you can leverage the fact that you're on a four-manifold to get an incredibly beautiful formulation of electrodynamics. IIRC (I don't have MTW in front of me) you write the Faraday tensor--actually a differential form--$F$, in terms of which Maxwell's equations become $$F^{ij}{;j} = J^i$$ and $F^{i[j}{;k]} = (dF)^{ij}_{k} = 0$, where I write $d$ for the exterior derivative. This can be further simplified by noting that the second implies (I think) that $F = dA$ for some 1-form $A$, the vector potential. – Christopher White Dec 07 '12 at 17:06
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Electricity and Magnetism for Mathematicians. A Guided Path from Maxwell's Equations to Yang–Mills is the only book I know that targets a mathematicial audience.

Urb
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Zythos
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