This book was designed for a two-semester sequence of courses on mathematical methods in the physical sciences and is aimed primarily at first year physics graduate students. A course in mathematical methods is often part of the physics curriculum, and it can be a challenge to both students and instructors.
If one were to write a book on carpentry that offered successive chapters on hammers, saws, drills, screws and nails, but never once addressed how to use the tools to make something, you would probably regard it as a failure. Yet there have been more than a few books like that on mathematical methods in physics. It can be difficult to get the balance right. There is a desire to “cover” as many concepts and techniques as possible, but without grounding and context, the whole effort can be a disappointment to both student and professor.
The current book has two distinct components. The first part of the book (Chapters 1 through 9) addresses more traditional mathematical methods. The second part (Chapters 10 through 19) considers modern differential geometry, topology, groups, group representations, and complex variables. All in all, there are a huge number of topics, but there are also many connecting links between methods and applications. The authors do a very creditable job of integrating the mathematics with the applications.
The exposition begins with the calculus of variations and the Euler-Lagrange equation; this leads immediately to good physical applications. The authors develop the idea of operators, beginning with matrices acting in finite dimensions and build on this to describe linear operators on function spaces. They then use the language of operators in their subsequent treatment of ordinary and partial differential equations and integral equations.
The second part begins with calculus on manifolds and the tools of the exterior calculus. Next comes some differential (and algebraic) topology, including homology and cohomology, characteristic classes and some Hodge theory. After that, there are chapters on groups and group representations, Lie groups and fiber bundles. The final three chapters address the basic elements of complex analysis as well as complex variables and special functions. Throughout the second half of the book there are strong links between mathematical ideas and physical applications, from gauge theory to supersymmetry.
The publisher’s blurb for this book says, “The author’s exposition avoids excessive rigor, whilst explaining the subtle but important points often glossed over in more elementary texts.” I suppose that “excessive rigor” is one of those things you recognize when you see it. It does not appear in this book. As far as I can see, there are no statements of theorems, and certainly no proofs. However, there is little here that a mathematician would find unacceptable. It is clearly a book written by physicists for physicists-in-training. For mathematicians, it offers some insight into how physicists think about and use mathematics. There are also plenty of nice applications and examples.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
Preface; 1. Calculus of variations; 2. Function spaces; 3. Linear ordinary differential equations; 4. Linear differential operators; 5. Green functions; 6. Partial differential equations; 7. The mathematics of real waves; 8. Special functions; 9. Integral equations; 10. Vectors and tensors; 11. Differential calculus on manifolds; 12. Integration on manifolds; 13. An introduction to differential topology; 14. Group and group representations; 15. Lie groups; 16. The geometry of fibre bundles; 17. Complex analysis I; 18. Applications of complex variables; 19. Special functions and complex variables; Appendixes; Reference; Index.