MAA Reviews

Tensors for Scientists: A Mathematical Introduction

Book cover for Tensors for Scientists: A Mathematical Introduction by Ana Cannas da Silva, Özlem Imamoğlu, and Alessandra Iozzi. The cover features a minimalist academic design with a pale gray-green background, thin horizontal white lines across the upper portion, and overlapping translucent geometric shapes. The title, “Tensors for Scientists,” appears in large dark green text, with the subtitle “A Mathematical Introduction” below in smaller black text. The authors’ names are listed near the top left. A dark green rectangular accent and a vertical red bar add contrast to the design. The publisher’s logo and name, Birkhäuser, appear at the bottom right.
  • Author: Ana Cannas da Silva, Özlem Imamoğlu, and Alessandra Iozzi
  • Series: Compact Textbooks in Mathematics
  • Publisher: Birkhäuser
  • Publication Date: 08/31/2025
  • Number of Pages: 178
  • Format: Paperback
  • Price: $49.99
  • ISBN: 978-3-031-94135-1
  • Category: textbook

[Reviewed by Geoffrey Dietz, on 06/19/2026]

As the title indicates, this book concerns tensors and is aimed at those who have not necessarily studied tensors before but hope to learn more about their uses in the physical sciences. By training, I am a commutative algebraist and have extensively studied tensors in the contexts of rings and modules. On the other hand, I have studied a decent amount of physics and have seen tensors in the context of mechanics and general relativity. Somehow, the two notions of tensors never fully harmonized in my mind, and so I was eager to read this book which seemed to have as its primary goal explaining tensors both in terms of pure algebra and of applications in physics.

This book is enjoyable to read and has a style (in the words of the authors) “between a textbook and lecture notes.” The authors’ skillful writing and presentation style made this book easy to read and to comprehend. Their writing is precise and engaging but also colloquial. It can be used as the main text for a course, but also works very well as an independent study guide. This book is one you can just pick up and read, take some notes if you wish, and work out the in-line exercises as you go.

The authors also clearly state their prerequisites at the start of the book. The potential reader should have completed the entire calculus sequence as well as an introductory course in linear algebra (covering vector spaces, change of bases, eigenvalues, and inner products). Most of the physics applications (moments of inertia, stress, strain, elasticity, and conductivity) do not arrive until the final chapter and show how tensors are used in practice. Although these concepts will be familiar to someone who has completed an introductory calculus-based physics course in mechanics, I think a person who has taken an intermediate mechanics course would be better positioned to appreciate these examples.

After a brief overview chapter, the authors review prerequisite linear algebra topics in Chapter 2 and introduce the Einstein tensor notation that is commonly used in the field. This chapter (and all the following chapters) includes lots of good examples and an assortment of exercises. (Solutions are provided at the back of the book.) The third chapter covers topics that might be found in a second course in linear algebra: dual spaces, linear functionals, bilinear forms, and the algebraic notion of a tensor product of vectors in an abstract vector space. This chapter is overall very well written except for the proofs of Propositions 3.10(2) and 3.27, which in their present form are unfortunately slightly incorrect.

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The fourth chapter covers inner product spaces, their connections to quadratic forms, and a review of the Gram-Schmidt process. The chapter closes with a nice section showing how the algebraic notions of covariance and contravariance relate to physics. I found two small typos in this chapter. In Example 4.8, the vector used in the calculations is listed with generic coordinates $v^1$ and $v^2$, but they should both be the number 1 based on the resulting calculations. In Example 4.29, the matrix G should have entries $g(b_i,b_j)$ instead of $g(b_i,b_i)$. Other than these minor issues, the chapter is again well-written and explains the material clearly.

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The fifth chapter explains general tensors in terms of multilinear forms so that one can understand tensors that have $p+q$ arguments where $p$ come from the dual space (contravariant arguments) and $q$ come from the original vector space (covariant arguments). They even discuss symmetric and antisymmetric tensors. Exercise 5.12 on this topic has a small typo in the solution. The dim of the space $S^2 V^*$ of symmetric 2-tensors is correctly given as $\frac{n(n+1)}{2}$ but is incorrectly labeled in the solution as $\binom{n}{2}$ instead of $\binom{n}{2} + n$. On the other hand, the dimension of $\Lambda^2 V^*$, the space of antisymmetric 2-tensors, is $\frac{n(n-1)}{2} = \binom{n}{2}$.

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So, although I pointed out a few minor typos and a couple of slightly incorrect proofs, I really enjoyed reading this book and picked up some extra insight into how the abstract algebraic notion of tensors relates to how physicists use them. If you wish to learn more about this area or to set up an inquisitive student with a book that can be read independently, I highly recommend this well-written and inviting text.

Geoffrey Dietz is a Professor of Mathematics at Gannon University in Erie, PA. He is married and has seven children.