Mathematical book on "Maxwell" equation

Question

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)

Answer 1

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.

Answer 2

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.

Answer 3

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)