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Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell’s Equations to Yang-Mills

Thomas A. Garrity
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
John D. Cook
, on

A mathematician might naively think that electricity and magnetism is not very interesting because the subject can be summarized in a system of linear differential equations. Such a perspective is narrowly reductionist. On the contrary, Thomas Garrity explains

… almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell’s equations. Even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries of the past thirty years. It seems that the mathematics behind Maxwell’s equations is endless.

Garrity’s new book Electricity and Magnetism for Mathematicians expands on his quotation above. The book is much broader than the title alone might imply. The subtitle A Guided Path from Maxwell’s Equations to Yang-Mills gives a better indication of what to expect. The book cuts through a large swath of modern mathematics and physics using electricity and magnetism as the theme that ties it all together. Electricity and Magnetism for Mathematicians seems to proceed at a leisurely pace until at some point the reader looks back to see that a great deal has gone by.

The book opens with a brief history of electricity and magnetism from the earliest understanding of these as separate phenomena, to their unification, and to sophisticated generalizations of their theory. It then introduces Maxwell’s equations in their differential and integral forms. Even though the final destination of the book is Yang-Mills theory, the book does not demand much background. It reviews basic vector calculus, for example. However, someone rusty on vector calculus is in for quite a ride by the end of the book.

Special relativity plays a major role in Electricity and Magnetism because electromagnetism and relativity are intimately related. Maxwell’s equations led to the discovery of special relativity. One can calculate the speed of light c from the electric constant ε0 and the magnetic constant μ0, i.e. \[ c^2 = \frac{1}{\varepsilon_0 \mu_0}.\] But wait a minute, this is the speed of light relative to what? Critics argued that Maxwell must be wrong because his equations lead to a constant speed of light without specifying the frame of reference. Einstein would come along later and argue that Maxwell’s equations were right but that our understanding of physics was wrong.

The connection between electromagnetism and relativity runs both ways. Electromagnetisms lead to the discovery of relativity, and relativity shows that electricity and magnetism are the same phenomena viewed from different perspectives. One way to state this is that magnetism is simply a relativistic effect of electricity and vice versa. As Garrity puts it

Coulomb’ Law + Special Relativity + Conservation of Charge = Magnetism

After an initial demonstration of the connection between relativity and electromagnetism, Garrity gives an introduction to Lagrangian mechanics and differential forms. This leads to a very satisfying expression of Maxwell’s equations in terms of the electromagnetic two-form. When stated in terms of the electric field E and magnetic field B, the symmetry between electricity and magnetism isn’t obvious.

\[ \begin{align*} \nabla \cdot E &=\frac{\rho}{\varepsilon_0} \\ \nabla \cdot B &= 0 \\ \nabla \times E &= -\frac{\partial B}{\partial t} \\ \nabla \times B &= \mu_0 \left(J + \varepsilon_0\right) \end{align*} \]

But when stated in terms of the electromagnetic two-form F and the current one-form J, Maxwell’s equations take on the satisfying form \[\begin{align*}dF &= 0 \\*d*\!F &= J\end{align*}\] where the stars denote the Hodge dual operator.

Next Garrity moves on to combining electromagnetism with quantum mechanics. This requires introducing a fair amount of physics—quantum mechanics—and mathematics—manifolds, vector fields, connections, and Hilbert spaces. This establishes one of the major themes of the book:

Force = Curvature

and leads to the book’s goal of the Yang-Mills equations.

Garrity includes a generous selection of exercises in Electricity and Magnetism even though the book is a brisk survey. This makes me want to take a look at his previous book Algebraic Geometry: A Problem Solving Approach which emphasizes exercises even more.

Electricity and Magnetism offers much more than a typical text on electromagnetism. (And to be fair, also much less. Not many electrical engineering homework problems here.) It uses electromagnetism as the thread running through a broad selection of topics in physics and mathematics.

John D. Cook is an independent consultant and blogs regularly at The Endeavour.

1. A brief history
2. Maxwell's equations
3. Electromagnetic waves
4. Special relativity
5. Mechanics and Maxwell's equations
6. Mechanics, Lagrangians, and the calculus of variations
7. Potentials
8. Lagrangians and electromagnetic forces
9. Differential forms
10. The Hodge * operator
11. The electromagnetic two-form
12. Some mathematics needed for quantum mechanics
13. Some quantum mechanical thinking
14. Quantum mechanics of harmonic oscillators
15. Quantizing Maxwell's equations
16. Manifolds
17. Vector bundles
18. Connections
19. Curvature
20. Maxwell via connections and curvature
21. The Lagrangian machine, Yang–Mills, and other forces.