An oft-encountered challenge in teaching a course dedicated to mathematical methods of physics is, to be blunt, making it feel like a physics course. While not all departments require or offer such a course, in the ones that do, the aim, whether at the undergraduate or graduate level, is typically to facilitate students' ability to see the majestic physics forest—when explored in their other classes—through the trees of mathematical techniques. Indeed, when studying physics, it can be all too easy for many students to get tangled up in the vines and underbrush of mathematical details, taking valuable time and mental energy away from learning physics concepts and developing their physics intuition. However, in a course focused primarily on mathematical methods, it can be difficult to maintain a close sense of connection to the physics. While all math method textbooks certainly contain numerous physical examples, most fall short of creating and sustaining such a connection consistently throughout.
When teaching our undergraduate course Mathematical Methods of Theoretical Physics, I had the pleasure of interacting for two semesters with authors (and brothers) Gary and Kenny Felder as I used draft versions of their book, Mathematical Methods in Engineering and Physics. With my own sister being a physicist as well, it gave me particular delight to work with another sibling-physicist pair. In my department, this one-semester course is required for physics majors and minors and aimed at the sophomore level, with the goal of preparing students with the mathematical foundations needed for the advanced undergraduate physics curriculum. I had previously taught the course using Arfken, Weber, and Harris as the text, following the choice of the instructor who preceded me. While Felder and Felder does not serve nearly as well as a doorstop, I found it much more suited to our undergraduate course than Arfken, Weber, and Harris in nearly every other way.
Felder and Felder have taken great care in crafting a truly pedagogical book aimed at students being introduced to the material for the first time. For more advanced students who are already familiar with the material in any given chapter, such as in a graduate math method course, the detailed explanations in Felder and Felder may be more than they want and a terser text such as Arfken, Weber, and Harris may be more suitable. As a practicing physicist, Arfken, Weber, and Harris (or the internet) is my go-to for quick reference to refresh my memory. But Felder and Felder is my book of choice for teaching undergraduates. It serves well not only the students being exposed to the material for the first time but also the students who have already learned to use certain mathematical techniques mechanically in order to solve problems but who have critical gaps in their understanding—complex numbers and Taylor series come immediately to my mind. The chapters are effectively independent, with prerequisites for any given chapter clearly listed at the beginning of the chapter; thus, it is easy to skip the material depending on the course content selected by the instructor. The topics range from the ones encountered and needed relatively early in the undergraduate physics curriculum, such as complex numbers, multivariable calculus, and ordinary differential equations, through to more advanced topics such as partial differential equations and complex analysis. Note that the book does not currently include any treatment of probability or statistics. Given that at my institution, physics students are only guaranteed to encounter probability and statistics in laboratory courses if they do not see them in Math Methods, I prepared the supplemental material myself to teach a brief unit on this topic, which was the only one for which Felder and Felder did not meet my needs.
I have not yet mentioned what I feel to be the greatest strength of the book: its tight connection to physical science throughout. This aspect, along with the carefully thought-out pedagogical explanations, sets it apart from Boas, which nominally has similar content and is aimed at a similar level. The “Motivating Exercises” at the beginning of each chapter are outstanding! Examples include vibrations in a crystal (for the chapter on “Taylor Series and Series Convergence”), the three-spring problem (“Linear Algebra”), flowing fluids (“Vector Calculus”), and the circular drum (“Special Functions and ODE Series Solutions”). I loved in particular the one from the chapter “Fourier Series and Transforms” on discovering extrasolar planets(!) and from the “Partial Differential Equations” chapter on the heat equation. I honestly never thought that I could be so enchanted by the heat equation before seeing how Felder and Felder effectively have students derive it as part of honing their intuition for how to think about partial differential equations. I have adapted the extrasolar planet motivating exercise as an exam problem to test Fourier analysis concepts when teaching the advanced undergraduate classical mechanics course. Felder and Felder also have “Discovery Exercises” at the beginning of most chapter sections, which guide students through deriving some of the key math ideas on their own before they are formally presented in the section. I recall watching the light bulb of understanding suddenly turn on above a number of my students' heads as they worked through the Discovery Exercise for the section on Maclaurin Series, in which they found a polynomial equivalent of the sine. Maclaurin and Taylor series had simply been recipes to them previously.
Felder and Felder offer further resources to students and instructors. There are “Check-Yourself” problems with answers provided and a wealth of excellent and varied problems (without answers in the back of the book) for each chapter, of which I made extensive use. While I did not include a computational component of the course when teaching it in the past, adopting the philosophy that this was perhaps the course more than any other in the undergraduate physics curriculum where students should dig into the underlying workings of mathematical techniques, I would consider adding a minor computational component in the future. To this end, I appreciate the handful of clearly marked, platform-independent computational problems provided for each chapter. Finally, there is yet further material available online at felderbooks.com, including exercises formatted for printing, extra problems for every chapter, answers to odd-numbered problems in the book, and even additional chapter sections with the corresponding problems.
Mathematical method texts are not intended to lay out the stately forest of physics as a whole to students or as a full exploration of any particular area. But rather than offering a catalogued description of many mathematical trees, Felder and Felder offer the unusual opportunity of an attentive guided walk through the forest, as they enthusiastically point out and explain the many and diverse roles that different mathematical techniques play in the larger physics ecosystem.
Christine A. Aidala is an associate professor in the Physics Department at the University of Michigan. Her research interests lie in experimental quantum chromodynamics and the internal structure of the proton.
