Elements of Tensor Calculus

This is a concise, semi-modern introduction to tensors, that is especially strong on applications. It is is suitable for math and physics majors. The present volume is a Dover 2016 unaltered reprint of the Wiley 1962 edition, which in turn was translated from the 1958 fourth French edition.

By “semi-modern” I mean that the mathematical approach is modern, using linear spaces and tensor products, while the notation is old-style, with lots of indices and differentials. The first 60% of the book is the theoretical part. It starts out with a brief introduction to linear spaces and manifolds, and then defines tensors in terms of tensor products. Tensors are not defined in terms of how they change under coordinate transformations, and in fact this introductory part is largely coordinate-free. The theoretical part is rounded out with a discussion of curvilinear coordinate systems and Riemannian geometry. The second 40% of the book is the applications, which are concise but cover a lot of ground. They include classical dynamics in tensor terms, dynamics and electromagnetics in special relativity (the longest part), and a little bit about gravitation in general relativity.

The biggest weakness of the book is that there are no exercises or worked examples. This lack would rule it out as a text for most courses today, although it would be a good supplement for a mathematical physics course. You can treat the applications portion as worked examples, but there’s no good way to jump ahead while working through the theoretical part. A good thoroughly-modern book, with lots of exercises and examples, is Neuenschwander’s Tensor Calculus for Physics.

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