Question

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 1

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.

Answer 2

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 3

• 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 4

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 5

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

Answer 6

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