This is a well-written and very traditional introduction to vectors as they are used in physics. The present volume is a Dover 2015 unaltered reprint of the 1963 Pergamon edition.

The book has clear but brief coverage of all the standard areas, including the basics of vectors, derivatives of vector functions, special vectors such as the gradient, integration of vectors, line integrals, surface integrals, and the Gauss and Stokes theorems. One topic you might expect, but that is omitted, is differential geometry. There is a very good final chapter on applications, that covers a wide variety of uses of vectors in geometry, mechanics, and a little bit of potential theory. Happily the book is much less concise in this chapter and goes into a great deal of detail on how these problems are worked out. The line drawings illustrating vectors in geometry and physics are very clear.

The exercises are especially good: numerous and difficult. Most of them were curated by the author from old British university examination papers, such as the Cambridge Tripos.

Most standalone books on vector analysis also include a lot of tensors; for example, Borisenko & Tarapov’s *Vector and Tensor Analysis with Applications*. The present book is unusual in that it sticks strictly to vectors. Its competition is not really other vector books but larger texts on mathematical methods. For example, Kreyszig’s *Advanced Engineering Mathematics* and Arfken & Weber & Harris’s *Mathematical Methods for Physicists* both have long sections on vector calculus that cover the same material as in the present book, and more. There aren’t many students who would need a separate text on vectors, because they would already be studying the same material as part of a more comprehensive methods course. The greatest value of the present book is in its exercises and in the chapter on applications.

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.