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Vector and Tensor Analysis with Applications

A. I. Borisenko and I. E. Tarapov
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a concise but thorough text in vectors and tensors from the physics (not linear algebra) point of view. The text is unusual in that tensors take central place. It starts out with vectors, but at least half the material in the rest of the book is stated for general tensors. It is one of a series of texts that Richard A. Silverman prepared in the 1960s and 1970s by translating and freely adapting Russian-language texts. The present volume is a Dover 1979 corrected reprint of the 1968 Prentice-Hall edition.

A good knowledge of physics is almost essential to use this book. The vector and tensor parts start from scratch, but all the examples are drawn from physics, with no explanation of the physical concepts used. The main areas of physics covered are fluid dynamics and electromagnetic theory. There is a great deal on the metric tensor but (oddly) nothing on relativity.

The applications come at the end of the book, in the last chapter, but are worth the wait. They are extended analyses of physics problems and use all the preceding material. I think the reason they come so late is that they make heavy use of vector calculus, and that is the last thing developed. There are also little examples scattered through the text, but most of these just point out that some physical quantity is a tensor without telling you how this information is useful.

Each chapter ends with a long section of solved problems and a brief list of exercises. The solved problems are difficult and are worked out in great detail. In some cases they extend the discussion in the text. The exercises tend to be drill or easy proof problems; most of them have the final answer in the text.

This is still a very good book for its intended audience. A good alternative is Neuenschwander’s Tensor Analysis for Physics; it is thoroughly modern and is better motivated than the present book. An oldie-but-goodie with much the same nature as the present book is G. E. Hay’s Vector and Tensor Analysis (Dover, 1953). It is not as thorough and in particular does not go very far in tensors, but may be more accessible because it draws most of its examples from mechanics and differential geometry.

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.

Editor's Preface

1. Vector Algebra

2. The Tensor Concept

3. Tensor Algebra

4. Vector and Tensor Analysis: Rudiments

5. Vector and Tensor Analysis: Ramifications