# Maxwell's equations and differential forms

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.

• Is space empty? Aug 5, 2011 at 10:35
• @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. Aug 5, 2011 at 14:57
• 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. Sep 25, 2017 at 14:27

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

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

• 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.

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.

• 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. Aug 6, 2011 at 11:47

• 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.

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.

• Under no circumstances should you enter to libgen, search for "Relativistic electrodynamics and differential geometry" and download the available pdf copy of this book.
– efs
Jun 14, 2021 at 2:25

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)

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

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

Stuck - On some mathematical aspects of deterministic classical electrodynamics

The article found through this link provides an exposition of all of electromagnetism using differential forms and exterior calculus.

All of Maxwell's equations are developed using differential forms from exterior calculus; undoubtedly all of this was known to Grassmann in his 1847 treatise on electromagnetism, but no one could understand both electromagnetism and exterior calculus.

I have had a go at explaining this at http://www.astro.uvic.ca/~jchapin/. Click on "E&M and the Faraday Field Tensor F".