What’s a tensor? This is a loaded question, and the answer is dependent on whom you ask. In pure mathematics, let’s say in (multi)linear algebra, a tensor is really presented as an element of a tensor product of, say, linear spaces, or something evolving from it. The key is that in the latter we have a construct possessing the property of universality in the sense that a multilinear mapping on a Cartesian product of vector spaces (or modules over a ring) can be canonically associated to a linear mapping on the corresponding tensor product of these spaces. Why and how? A partial answer to the first question is that dividing out by multilinearity (which is how one constructs a tensor product) is in and of itself desirable since it makes computations very much simpler, and one gains the benefit of being able to use the ever so powerful results of linear algebra directly. The second question’s answer is already hinted at in the preceding comment: it’s about dividing out by an ideal (natural in hindsight) generated by the relations engendering multilinearity. The corresponding theorems are a great deal of fun to prove; see, for example, Keith Conrad’s excellent notes, https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf and https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod2.pdf .

Getting down to a few brass tacks, if we look at finite dimensional cases and assign bases, so that linear mappings look like matrices, what happens is that the tensor product takes on the shape of an object with a famous name, the Kronecker product (of matrices), and we have the wherewithal to get pretty explicit early on. More generally, and as already observed above, just as a vector is just an element of a vector space, a tensor is an element of a tensor product, and the above construction, dividing out by linearity, succeeds in providing induced algebraic structure to these individual tensors; more about this in a moment.

Staying in pure mathematics for now, it is proper to note that in addition to the appearance of the tensor product in computational algebra, this beast is ubiquitous in differential geometry, e.g. in the context of differential forms on a suitable manifold (cf. the definition of the exterior algebra). More importantly for present purposes, however, in differential geometry or, specifically, in Riemannian geometry, one begins to encounter honest-to-goodness tensors in very suggestive ways, e.g. as measures of curvature. Here we are dealing with a shift in perspective really: tensors are now front-and-center as meaningful computational objects, instead of the earlier tensor products of spaces or modules, even if these are necessarily present in the shadows. These tensors come in two flavors at the outset, covariant and contravariant, and this distinction is reflected in a universally accepted notation: lower indices for covariance, upper ones for contravariance. And so it is that a very slick yoga of mixed tensors can be delineated, and this has proven to be of great utility in geometry.

But the book under review is a physics book: it’s part of Springer Verlag’s Undergraduate Lecture Notes in Physics series, so it’s not geometry. Or is it? Well, of course it is, in a way: Riemannian geometry was famously Einstein’s choice as the vehicle par excellence for general relativity, and that set the ball rolling. Shapiro’s book emphasizes this connection explicitly in its very title: the reader is to learn tensor analysis as the physicists do it in the context of relativity. His approach is three-pronged; we’ll let him speak for himself: “… tensor calculus is necessary for successfully learning all parts of theoretical physics, and … it is better to learn tensors in the second year of an undergraduate program … after learning linear algebra and calculus of several variables. When I was a student … in Russia, we had such a course … The first part of the present book is an essentially extended version of th[is] tensor course … The main point of this part is that the geometric (and consequently) physical laws should be formulated in such a way that the dependence on the choice of coordinates can be kept under control.”

“The second part … is intermediate … Certainly, a physicist has to know much more about electrodynamics that it is possible to learn from Part II of this book. However, our aim is not to replace the Electrodynamics courses, but rather to show intermediate-level applications of tensor analysis.” Here we predictably encounter Maxwell, of course, and special relativity.

Finally, “[t]he third part of the book [concerns] general relativity … The organization and contents of this part are a little bit different from the standard courses, mainly because they include material most useful for … students who are going to work in quantum field theory and quantum gravity.”

Thus, Shapiro’s book is a promising player in the game of physics texts aimed at the modern student with correspondingly modern interests. Beyond this, especially given, e.g., the role played by quantum field theory in various contemporary mathematical pursuits (just consider low-dimensional topology and the topological quantum field theory of Witten, Atiyah, Segal, for example), this book has additional mathematical relevance.

It is a very solid pedagogical effort, with the various physical themes all carefully worked out, as also the relevant mathematics, of course. Perhaps most importantly, there are many, many problems, and, as always, the student’s mastery of this material is proportional to the number of problems worked.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.