This is a very well motivated introduction to tensors, aimed at upper-division undergraduate physics majors. The author begins by admitting that the usual definition of tensors in terms of invariance under change of coordinates is very mysterious, and instead of starting there he works up to it gradually with a series of examples from physics (mechanics, electrodynamics, and special relativity). He starts with vectors, pointing out that coordinate systems are a human invention and that the physical quantities involved must be invariant regardless of the type and position of the coordinate systems. Coordinates are how we measure but Nature doesn’t know anything about them. From this he gets that the physical quantities must be invariant under all kinds of coordinate changes (including but not limited to the usual rotations) and shows how this can be abstracted to define vectors.

He then considers several physical problems in which the equations include quantities that turn out to be rank-2 tensors. The older notation of dyads is covered briefly but not used for anything. There is a great deal on special and general relativity as part of the motivation, but it’s not comprehensive.

The exercises are very strong. They are divided into discussion questions, which often ask “why do we do it this way and not that way?” to test your understanding of the development, and actual exercises, a few of which are drill but most are to prove something or calculate some important formula.

Most of the book works by example and analogy and is not very formal. It develops general rules and properties of tensors, but the emphasis is on specific physical applications. The last two chapters revisit the material from a more formal and mathematical perspective, covering manifolds and differential and multilinear forms.

One weakness of the book, from a more advanced standpoint, is that it only studies tensors that are already known to be tensors; there’s no guidance about discovering or creating tensors. Historically tensors came out of differential geometry, and were already well-understood by the time Einstein needed them for general relativity. His approach was geometric, and he knew from the kind of geometries he was considering that he would need tensors. The present book works more from known formulas where the physical laws have already been discovered, and it is not very geometrical. The material on differential forms gives a different formalism for what has already been covered, and is not approached from geometry.

Bottom line: a very good book for physics students. It’s good for math students too if they have a strong physics background (otherwise the examples are too hard to follow).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.