Maxwells equations and differential forms - MathOverflow

Maxwells equations and differential forms

2011-08-05T09:09:47Z

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

Answer by Ben McKay for Maxwells equations and differential forms

2011-08-05T10:30:26Z

Bernard F. Schutz, Geometrical methods of mathematical physics, p 175, chapter 5.11 Rewriting Maxwell's equations using differential forms.

Answer by Cristi Stoica for Maxwells equations and differential forms

2011-08-05T11:07:57Z

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

Answer by Fedor Petrov for Maxwells equations and differential forms

2011-08-05T12:15:22Z

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

Answer by Qfwfq for Maxwells equations and differential forms

2011-08-05T12:37:18Z

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.

Answer by Stefan Behrens for Maxwells equations and differential forms

2011-08-05T13:27:49Z

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.

Answer by Paul Siegel for Maxwells equations and differential forms

2011-08-05T15:35:19Z

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.