An introductory Tensor Calculus for Physics book is a most welcome addition to the libraries of both young students in physics and instructors with teaching duties at the advanced undergraduate level. Indeed, the literature on the subject, notwithstanding how ample it is, lacks books that are both at an introductory level and have young physicists as a preferred audience. Professor Neuenschwander's book fills the gap in robust fashion.

The book comprises eight chapters and may be ideally divided into two parts, with the first five chapters containing the core of the subject. Besides reviewing some basics in vector calculus, Chapter 1 explains very clearly how the need for physical quantities to have a tensor character arises. In this way, rather than defining tensors as multicomponent entities with a specific transformation law under a coordinate transformation, the concept is very appropriately introduced as a necessary requirement to represent physically meaningful quantities. The chapter is probably longer than what would be necessary, and some of the examples presented with a clarifying aim might not be a best choice (see, e.g., the example of the transformation between inertial frames of the electric and magnetic fields in Sec. 1.8), yet it is a good starting point for readers who would use the book as a self-study tool. Moreover, the exercises at the end of the chapter have been very well chosen for the intended level.

With enough motivation from Chapter 1, Chapter 2 considers some of the higher-rank tensors that undergraduate students might have encountered in their introductory courses: the inertia tensor from mechanics courses and the electric susceptibility, the electric-quadrupole, and the electromagnetic-stress tensor from E&M courses. Regrettably, the author omits the deformation tensor arising from applying a stress to solid bodies: the three-dimensional deformation of a rubber band when it is stretched along a single direction is perhaps a very effective example of the emergence of higher-rank quantities. Again, the exercise collection is well chosen, although here some of the exercises are more advanced than expected (see, e.g., ex. 2.17, which requires a non-trivial familiarity with quantum mechanics).

In Chapter 3 and 4, the author does a superb job in presenting the all important metric tensor, to which a detailed treatment is devoted, and the covariant derivative, quite well motivated and explained. Both chapters are equipped with very doable and instructive exercises. Finally, the last chapter of the core of the subject prepares the path for applying the theory to general relativity, a step taken in Chapter 6, where the basic theory is applied to electrodynamics as well, at a level well weighted for the intended audience. An excellent chapter is the last one (Chapter 8), where the reader is beautifully and gently introduced to the more advanced differential forms, which is how the subject should be eventually mastered when the readers reach a higher level.

Chapter 7 puzzles us in that the material covered there, especially the dual vector space and dual vector bases, could have fit perhaps better in a chapter between the first and the second. The idea that a scalar product is performed between two vectors belonging to different spaces, one dual to the other, should be conveyed as early as possible, together with the clarification of self-duality of Euclidean space parameterized by an orthonormal basis.

Given the level of the book, having solutions to the exercises at the end would have been helpful for the reader, as well as comments on the discussion questions that the author poses at the end of each chapter.

As a last remark, the author might consider in a subsequent edition of the book the use of Schouten's notation, according to which the primes denoting the “primed” coordinate frame are attached to the component index rather than to the kernel letter denoting a multicomponent quantity, say Ak rather than Ak. This is a matter of taste, but our experience suggests that Schouten's notation is much more effective when, upon performing algebra where several multi-indexed quantities are involved, keeping track of the indices might become somewhat frustrating, especially for the novice.

Apart from some minor criticisms, we think that the book, which starts from the very basic and goes on gently up to fairly advanced material, deserves the best attention by both young physicists and instructors.

Franco Battaglia is professor of chemical physics at the Department of Engineering Enzo Ferrari, University of Modena, Italy. He has been active in several fields of theoretical chemical physics: molecular scattering theory, atom-surface interactions, phase transitions in 2D systems, and many-body theory. He has coauthored two books in classical and quantum physics and in chemical physics. In collaboration with Professor T. F. George, he has proposed a guideline (published in AJP in 2013) for effective teaching of tensor calculus at the advanced undergraduate level.

Thomas F. George has served as chancellor at the University of Missouri-St. Louis since 2003, and prior to that he was chancellor for seven years at the University of Wisconsin-Stevens Point. As professor of chemistry and physics, he maintains an active research program in chemical/materials/laser nanophysics, including nanomedicine. His work has led to 760 papers, 5 authored textbooks, and 18 edited books/volumes. He is also a jazz pianist performing throughout the St. Louis region, and his performances have extended to Illinois, Arkansas, and overseas to China, Croatia, Hungary, Kuwait, Romania, and Serbia in connection with his travels as chancellor and scientist.