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.
|Publisher:||Cambridge University Press|
|Product dimensions:||5.98(w) x 8.98(h) x 0.91(d)|
About the Author
Thomas A. Garrity is the William R. Kenan, Jr Professor of Mathematics at Williams College, where he was the director of the Williams College Project for Effective Teaching for many years. He has written a number of research papers and has authored or coauthored two other books, All the Mathematics You Missed (But Need to Know for Graduate School) and Algebraic Geometry: A Problem Solving Approach. Among his awards and honors is the MAA Deborah and Franklin Tepper Haimo Award for outstanding college or university teaching.
Table of Contents1. 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.