# Mathematical book on "Maxwell" equation

I am looking for a classical book on Maxwells equation. Let me explain: For potential theory I found "Foundations of Potential Theory from Oliver Dimon Kellog from 1929. It is basically a mathematical book on electrostatics. Slow, detailed, and existence proof are about 'normal' function (not distributions, of course). I found this, because somebody mentioned this area is called 'potential-theory'

I am now looking for something similiar on either (time dependent) Maxwell-Equations or wave-equations. Of course the modern term would be partial differential equations; but I am looking for a classical text about uniqueness/existence proof all in the real of 'classical' mathematics,so pre 1950.

I am sure there must have been good books on that topic, since physics has been a driving force for mathematics, but I dont seem to be able to find them.

Edit: What I might be looking for is a classical (pre-Distribution) text on hyperbolic differential equations. Although not too sure if this is the right term (finding the term 'Potential theory' took quite some time)

• You might be interested in reading Integral Equation Methods in Scattering Theory by Colton and Kress. It is a mathematically rigorous treatise on Integral Equations for Electromagnetics. Jun 25, 2018 at 20:05
• zbmath.org/?format=complete&q=an:0588.35003 Jun 25, 2018 at 20:43
• Uniqueness? Are you sure Maxwell's choice of fields (scalar and vector) to underlie electromagnetism is unique? Jun 25, 2018 at 23:13
• @JamesArathoon I am not sure, but if you set a sufficent set of boundary condition then the solution to the Maxwell equation should be unique, or shouldnt it? Jun 26, 2018 at 7:49
• Yes as long as you only worry about the interaction energy or free energy components in the field. For example Maxwell never solved the problem of particle self-energy, and charged particle stability within his theory. This energy has to exist (I personally think distributed in the field in a unique way) but it is not pinned down in a unique way in Maxwell's Theory. This problem hasn't been properly cleared up yet and is partly why we end up with a patch work of theories in physics, each "successfully" valid in its own limited domain. Jun 26, 2018 at 11:33

Below are some books arranged chronologically that I either already knew about ([1], [2], [3]) or found just now by googling. In some ways, [1] is closest to Kellogg’s book (a 1979 roommate of mine, and friend during my undergraduate years, worked through most of Kellogg’s book during 1978-79 at the suggestion of one of our professors), but Jeans was a physicist whereas Kellogg was a mathematician, and thus as each book progresses, Kellogg’s book focuses on mathematical issues much more than Jeans’ book. Sommerfeld’s book [2] was often cited and recommended when I was still somewhat involved in physics (late 1970s to early 1980s), and it certainly fits the bill as a pre-distribution book, but like Jeans, Sommerfeld was also a physicist, so you have to not take some things too mathematically literally. For example, the absolute value function would be said to have a discontinuity in its derivative (because its derivative is $-1$ for $x<0$ and $+1$ for $x>0).$ Thirring’s book is probably too advanced, but I felt it shouldn’t be omitted in a list such as this. I don’t know much about the other books, but since I also don’t know anything about your background, I figured that each was worth including.

[1] James H. Jeans, The Mathematical Theory of Electricity and Magnetism (1927; latest changed edition, freely available)

[2] Arnold Sommerfeld, Partial Differential Equations in Physics (1949; freely available)

[3] Walter Thirring, Classical Field Theory (1st edition 1979, 2nd edition 1986)

[4] Cessenat Michel, Mathematical Methods in Electromagnetism. Linear Theory and Applications (1996)

[5] Piero Bassanini and Alan Elcrat, Mathematical Theory of Electromagnetism (2009; freely available)

[6] Kurt O. Friedrichs, Mathematical Methods of Electromagnetic Theory (2014)

[7] Thomas A. Garrity, Electricity and Magnetism for Mathematicians (2015)

• Thanks for this answer! So which one of these books actually proofs any theorems? If read [2], and as usual Sommerfeld is great, but theorem free. [3] is great as well, a few time the word 'proof' is used, but these might only be sufficient for a mathematician already well versed in the subject. [1] is the closest (as far I can tell), could it be, that not so much was know about wave equations before distribution theory? Jun 30, 2018 at 17:11
• My impression is that [4] through [7] are fairly mathematical, but besides [5] (freely available on the internet), I haven't seen any of the last 4 books and thus I don't know anything more than you know (after some superficial googling for reviews). Incidentally, you can do a more thorough search for reviews of books (in general, not just these books) by a JSTOR Advanced Search. Select "All Content", select "Reviews", and in one blank enter the first and last name of one author (no quotes used) and in the other blank (continued) Jul 1, 2018 at 10:34
• enter three or four words of the title between quotes, such as "classical field theory". Unless you have JSTOR access, you will not be able to read the reviews, but at least you will know of the existence of any such reviews, and whether it would be worth your time/effort to visit a university library at some later time to attempt to locate the reviews in order to read them. Jul 1, 2018 at 10:37
• Incidentally, for a couple of rather rigorous books on elementary/intermediate level classical mechanics, see my comments here. The fact that one is written by Osgood and the other is written by Banach should be all I need to say about their rigor! Of course, these are not intended to be mathematically oriented books, but given the mathematical renown of these two, I think they're definitely worth adding to your digital book collection (both are freely available). Jul 1, 2018 at 10:50

I think that the best book of such kind is the monograph by Claus Müller (1969) [1], which is the translation of an older 1957 monograph: first of all, the Author was the mathematical physicist who proved the existence of the solution to the problem of diffraction of electromagnetic waves by a dielectric ball, together with Hermann Weyl and Victor D. Kupradze.

The author does not use distribution theory: by using the integral definition of standard vector operators, he defines a sort of weak derivatives in the following way: $$\begin{split} \nabla\cdot \boldsymbol{a}&=\lim_{G_i\to x}\frac{1}{\Vert G_i\Vert}\int\limits_{\partial{G_i}} \boldsymbol{n}\cdot \boldsymbol{a}\,\mathrm{d}\sigma\\ \nabla\times \boldsymbol{a}&=\lim_{G_i\to x}\frac{1}{\Vert G_i\Vert}\int\limits_{\partial{G_i}} \boldsymbol{n}\times \boldsymbol{a}\,\mathrm{d}\sigma\\ \nabla\varphi&=\lim_{G_i\to x}\frac{1}{\Vert G_i\Vert}\int\limits_{\partial{G_i}} \boldsymbol{n}\varphi\,\mathrm{d}\sigma\\ \end{split}\tag{1}\label{1}$$ where

• $$\boldsymbol{a}$$ is a (non-differentiable) vector field in $$\mathbb{R}^3$$,
• $$\{G_n\}$$ is an contractible indexed family of smooth sets in $$\mathbb{R}^3$$ converging to the point $$x\in\mathbb{R}^3$$, whose volume is $$\Vert G_n\Vert$$ and whose boundary surface is $$\partial{G}_n$$,
• $$\boldsymbol{n}$$ is the inward normal vector to the surface $$\partial{G}_n$$.
• $$\varphi$$ is a (non-differentiable) vector field in $$\mathbb{R}^3$$.

The entire first chapter is devoted to the development of the vector analysis for the "generalized operators" defined by \eqref{1}. After having introduced the standard spherical harmonics and Bessel functions in chapter 2, by using the vector analysis defined in the first chapter, the Author studies Maxwell equations by using time-harmonics waves, i.e by transforming them to properly defined reduced wave (Helmholtz) equations $$\begin{split} \Delta\varphi+\varphi&=0 \text{ (scalar field)}\\ \Delta\boldsymbol{a}+\boldsymbol{a}&=\boldsymbol{0}\text{ (vector field)} \end{split}\tag{2}\label{2}$$ Then the study follows by developing the theory of EM waves in homogeneous and inhomogeneous media, and boundary value problems for equations of type \eqref{2}. Finally, you can find more details on its contents by reading the long ZBMATH review of the 1957 edition.

Edit. I noted the OP later EDIT and felt to add the following observation. It is not customary to develop the theory of Maxwell Equations in the time domain, so if you are searching about information on hyperbolic equations in a text on electromagnetism, probably you will not find what you are looking for. It is customary in the electromagnetic theory to study the so called Helmholtz equation which can be deduced from the wave equation by using time-harmonic EM fields, i.e. fields of the form $$\boldsymbol{A}(x,t)=\boldsymbol{A}_0(x,\omega)e^{-i\omega t}$$ The Helmoltz equation is elliptic, and essentially the authors who if you adopt this choice develop the 'Potential theory' of this equation.

[1] Claus Müller (1969)[1957], Foundations of the Mathematical Theory of Electromagnetic Waves, Grundlehren der mathematischen Wissenschaften 155, Springer-Verlag, pp. VIII+353, MR0253638, Zbl 0181.57203.

• This looks very promising. I will try to get the book asap. Your hint that things are done using Helmholtz is very good. Although for.uniqueness of the wave solution depending speficyinf initialconditions on a surface in the backward lightcone- can this be done using Helmholtz? Jul 3, 2018 at 6:48
• Yes, because the use of time-harmonic fields basically is equivalent to applying the method of separation of variables: the spatial part is determined by using the Helmholtz equation, which accounts for Dirichlet, Neumann, Robin, impedance or whatsoever boundary condition, on the interior or on the exterior of a given (even multipli connected) domain. Then, the initial condition is accounted by performing some kind of (generalized) Fourier analysis, i.e. $$\boldsymbol{A}(x,t)=\int\limits^{+\infty}_{-\infty} \boldsymbol{A}_0(x,\omega)e^{-i\omega t}\mathrm{d}\omega$$ Jul 3, 2018 at 8:40
• @lalala. Beware: there is a typo at page VII of the book (in the preface). You'll find it by yourself. Also the author does not fully uses the standard notation of equation \eqref{1} above: the divergence operator of a vector field is indicated as "$\nabla\boldsymbol{a}$, i.e. using the notation for the gradient applied to a vector field instead of to a scalar field (he drops the "$\cdot$"). However the notation is coherent and the Author fine exposition avoid any confusion. Jul 3, 2018 at 8:50
• @lalala yes, the deduction and validity of Helmholtz's equation implies the applicability of the superposition principle to time-harmonic fields, thus this approach is intrinsically linear. Dec 18, 2020 at 18:49
• @lalala it is mostly the second reason you quoted but not entirely. Nearly all the general theorems on the EM-fileds can be proved both in the linear and nonlinear setting. And also there are very interesting accounts of the theory of EM-fields in nonlinear media, as are the works of Frederick Bloom, Eringen & Maugin, Kunin and others: the matter is however difficult and it is unclear if some of the remaining open problems (also for the linear theory) can be settled by the existing techniques, thus perhaps many mathematicians prefer to investigate something that guarantees a job. Dec 19, 2020 at 15:17

I like Feynman's chapter on Maxwell's equations:

http://www.feynmanlectures.caltech.edu/II_18.html

Googling "Maxwell's equations for dummies" also comes up with a number of sites.

• Well, I like Feynman as well, but I am looking for a math (not physics) text. E.g. Feynman doesnt specify the what functions he is using (maybe C^infinity intersected with L^2 ???) and so on and on. The approach is usually quite different. Jun 26, 2018 at 7:51