Electricity and Magnetism for Mathematicians
A Guided Path from Maxwell's Equations to Yang–Mills

Maxwell's equations have led to many important mathematical discoveries. This text introduces mathematics students to some of their wonders.

Thomas A. Garrity (Author)

9781107435162, Cambridge University Press

Paperback / softback, published 19 January 2015

294 pages, 81 b/w illus. 2 colour illus. 211 exercises
22.8 x 15.1 x 1.5 cm, 0.4 kg

This text is an introduction to some of the mathematical wonders of Maxwell's equations. These equations led to the prediction of radio waves, the realization that light is a type of electromagnetic wave, and the discovery of the special theory of relativity. In fact, 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. The goal of this book is to explain to mathematicians the underlying physics behind electricity and magnetism and to show their connections to mathematics. Starting with Maxwell's equations, the reader is led to such topics as the special theory of relativity, differential forms, quantum mechanics, manifolds, tangent bundles, connections, and curvature.

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.

Subject Areas: Mathematical physics [PHU], Electricity, electromagnetism & magnetism [PHK], Applied mathematics [PBW], Topology [PBP]