# Maxwells equations and differential forms

Hi,

is there a textbooks that explains the maxwell equations in differential form?

What I understood so far is, that the $E$ and $B$ fields can be assembled to a differential 2 Form $F$, and the Maxwell Equations can be written quite nicely with the Hodge $*$ and the exterior deriative $d$. Going further the equations can be derived as an Euler Lagrange (or Yang Mills?) equation 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.

Thanks

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Is space empty? –  Charles Matthews Aug 5 '11 at 10:35
Hoping to be useful to you, I added the reference-request'' tag. –  Giuseppe Tortorella Aug 5 '11 at 14:20
@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. –  José Figueroa-O'Farrill Aug 5 '11 at 14:57

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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Here is a link to my book mentioned above: vmm.math.uci.edu/GeometrizationOfPhysics.pdf –  Dick Palais Aug 6 '11 at 16:35

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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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. –  Spiro Karigiannis Aug 6 '11 at 11:47

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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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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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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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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