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Is there a textbook that explains Maxwell's equations in differential forms?

What I understood so far is that the $E$ and $B$ fields can be assembled to a 2-form $F$, and Maxwell's equations can be written quite nicely with the Hodge $*$ and the exterior deriative $d$. Going further the equations can be derived as Euler-Lagrange (or Yang-Mills?) equations from a connection of a fibre bundle.

I am searching for a book that describes how the geometric entities are mapped to the physical entities with a focus on mathematical exactness.

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  • $\begingroup$ Is space empty? $\endgroup$ – Charles Matthews Aug 5 '11 at 10:35
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    $\begingroup$ @Charles: I don't understand the question. The Maxwell equations with sources also admit a differential form formulation. For example, in the absence of magnetic sources, they are $dF = 0$ and $d\star F = \star J$, where $J$ is the source's electric current. $\endgroup$ – José Figueroa-O'Farrill Aug 5 '11 at 14:57
  • $\begingroup$ If you wish something shorter, I wrote a paper: Non-linear electromagnetism and special relativity. Discrete Cont. Dynam. Syst., 23 (2009), pp 435-454. It treats also the nonlinear case. $\endgroup$ – Denis Serre Sep 25 '17 at 14:27
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Bernard F. Schutz, Geometrical methods of mathematical physics, p 175, chapter 5.11 Rewriting Maxwell's equations using differential forms.

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Baez & Muniain, Gauge Fields, Knots and Gravity, chapter 5, p. 69, Rewriting Maxwell's equations.

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  • M. Nakahara, Geometry, topology and physics. Paragraph 10.5 "Gauge theories", specifically 10.5.1 "$U(1)$ gauge theories".

  • R.S. Palais, The geometrization of physics, lecture notes from a course at National Tsing Hua University Hsinchu, Taiwan June-July 1981 [available on the internet, I think] Specifically, the paragraph "Generalized Maxwell equations"

  • G.Svetlychny, Preparation to gauge theory [freely available on the ArXiv]. Chapter 7 "electromagnetism" paragraph 7.1 "Maxwell's Equations".

  • H.Youk, A survey on gauge theory and Yang-Mills equations [available on the internet, I think]. Paragraph 7.1 "The Hodge-star operator and Maxwell's equations".

  • Eguchi et al., Gravitation, gauge theories and differential geometry.

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I really strongly recommend chapter 2 of Naber's "Topology, Geometry, and Gauge Fields: Interactions". In this book and its companion volume "Topology, Geometry, and Gauge Fields: Foundations", Naber provides a detailed, self-contained introduction to topology and geometry with a view toward physics. Both books are extremely detailed and strike an excellent balance between mathematical sophistication and physical motivation (with a bias toward the math). I can't recommend them enough.

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  • $\begingroup$ I agree. These are the books I always point to when undergraduates ask me where they can start to learn about more advanced topics in geometry, with an emphasis on relations to physics. They are very clearly written, too. $\endgroup$ – Spiro Karigiannis Aug 6 '11 at 11:47
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An important reference to add the already good answers is the two volume set:

  • Paul Bamberg, Shlomo Sternberg "A course in mathematics for students in physics" (Vol 1, Vol 2)

It is based on a Harvard course given by the authors back in the 80's, and it is basically a book on the calculus of differential forms geared towards physical applications: gaussian optics, electrical networks, electrostatics, magnetostatics, Maxwell's equations, thermodynamics are some of the topics discussed in the book in this setting.

Chapter 19 on Volume 2 is exactly about Maxwell's equations in differential form.

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Bolibruch's nice explanation is here: www.mccme.ru/free-books/dubna/bol1.pdf I do not know whether it has English translation, sorry.

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I remember that when I was studying for an exam in electrodynamics I really liked this book:

  • Parrott, Stephen: "Relativistic electrodynamics and differential geometry" (Springer, 1987) (MathSciNet Review)

It aims to give a mathematically precise treatment of the fundamentals of classical electrodynamics in the language of Lorentzian geometry. Unfortunately, it seems to be difficult to come by. Neither Google books nor Amazon have a preview for it.

But it's really worth a look, not only because of the mathematical language, but because it discusses something that usually gets swept under the rug, namely that a charged "test particle" in an electromagnetic field is not only affected by the field but actually interacts with the field! This has always bugged me in my physics courses.

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The best exposition that I am aware of is in Section 2.9 of Lewis Ryder's book "Quantum Field Theory":

http://www.amazon.com/Quantum-Field-Theory-Lewis-Ryder/dp/0521478146

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I would add:

  1. http://em.groups.et.byu.net/pdfs/publications/formsj.pdf (K. F. Warnick, R. H. Selfridge and D. V. Arnold, Teaching Electromagnetic Field Theory Using Differential Forms)

  2. http://link.springer.com/book/10.1007/978-1-4612-0051-2/page/1 (F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics)

  3. http://www.amazon.com/Classical-Electrodynamics-Roman-S-Ingarden/dp/8301053429 (R. S. Ingarden and A. Jamiolkowski, Classical Electrodynamics)

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