Frustrated by the treatments of tensor calculus in relativity books, I turned to this book and was not disappointed - it gets the job done in a logical, concise and admirably clear manner. I was skeptical at first as I like to understand things algebraically and this book is all about the traditional components based approach. But I've become a convert since this is what one needs to understand those tensor-based relativity books and as I discovered much to my chagrin, one can understand vector spaces, their duals, and multilinear functions till those cows come home without gaining much insight or any proficiency with all those tensor equations decorating relativity books.
Consider this: the book has 13 chapters, whose collective page total is about 213 pages but excluding the Solved Problems is less than 100 pages. So excluding pages devoted to solved problems, exercises, etc. the chapters look like this.
-- Chs 1 & 2 provide about 8 pp. of mathematical preliminaries (Einstein summation convention and some linear algebra).
-- Ch 3 defines and elucidates General Tensors, zipping you through the necessary details of coordinate transformations, the Jacobian matrix and Jacobian, the contravariant / covariant topic (minus the algebraic explanation, unfortunately), includes a nice section on Invariants (only p. 28, mind you), and ends with the Stress Tensor and Cartesian tensors, all in about 10 pages!
-- Ch 4 covers the basics of tensor algebra and tests for tensor character, in a mere 4 pages. For me those first 4 chapters were the painful part but really it was only about 23 pp.
After that things really picked up because the topics became more interesting:
-- Ch. 5 (8 pp: the Metric Tensor;
-- Ch. 6 (8 pp): the Derivative of a Tensor;
-- Ch. 7 (7 pp): basic Riemannian Geometry of Curves;
-- Ch 8 (6 pp): Riemannian Curvature, including the Ricci tensor (!!);
-- Ch 9 (6 pp): Spaces of Constant Curvature including the Einstein tensor (!!);
-- Ch 10 (12 pp.): Tensors in Euclidean geometry; and
-- Ch 12 (10 pp): Tensors in Special Relativity (!!).
I found Ch 12 to be a concise, lucid discussion of some essential aspects of special relativity from a tensor point of view.
-- Ch 11 (5 pp) deals with Tensors in Classical Mechanics, which I only skimmed quickly.
-- Ch 13 (12 pp), the final chapter, provides a brief introduction to tensor fields on manifolds (aka the modern approach) and is, I think, the weakest, least helpful chapter. Section 13.5 Tensors on Vector Spaces, in particular, struck me as way too short for such a central topic. Having studied the material in this chapter elsewhere, I find it hard to believe one could really understand the material from such a brief overview. But at least you can see "what you're up against". For this material I thnk one needs to study a differential geometry book such as Tu's lucid and concise An Introduction to Manifolds (Universitext), John Lee's long but self-study friendly Introduction to Smooth Manifolds, Jeffrey Lee's new Manifolds and Differential Geometry (Graduate Studies in Mathematics) (includes fiber bundles) or even Bishop and Goldberg's classic Tensor Analysis on Manifolds. However, I found Bishop & Goldberg to be a bit dated and a bit too concise, except as a review / consolidation of what I'd learned elsewhere (but it's superbly written and well worth reading!).
Of course, if you want examples and solved problems, Kay's book has plenty: and let's face it, the only way to acquire an intuitive feel for tensor equations or become remotely facile in tensor operations is through examples and (solved) problems. When I first read the book (a bit too quickly), I skipped many of these but on reviewing the material, I have come to appreciate them.
So initially I thought Kay's book was a poor choice (boring, too applied, too elementary) but having gained more experience, I have come to see that this book, although not perfect (what a surprise!), really is one very good - and economical - book on tensor calculus, both geared to self-study and especially well-suited for relativity enthusiasts.
Lastly, here are two books on tensors that I found to be unhelpful for relativity studies: A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) and Introduction to Tensor Calculus and Continuum Mechanics although they might suit those interested in continuum mechanics.