Schaums Outline of Tensor Calculus

You better know geometry, trigonometry and differential and integral calculus cold before getting this book. But if you do and you can interpolate across a few missing steps, it is great preparation for a book on General Relativity.

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.

This 1988 book by David Clifford Kay is packed with practical down-to-Earth no-nonsense tensor calculus for Euclidean, Riemannian and (flat) pseudo-Riemannian differential geometry. It's all applicable. There's no philosophical mumbo-jumbo about horizontal sub-bundles on principal bundles, the Hopf-Rinow theorem, holonomy groups, homotopy, homology, Hausdorff separation, or just about anything else that starts with H. But the extent of coverage and the level of detail are impressive, especially considering that the theory pages are dwarfed by the problem pages, which are fully solved! So this is an active learning book for a mathematical subject which is generally considered to be more painful than most. I wish I had started my own study of DG with this book. I would definitely recommend it to anyone who needs tensor calculus for applications to other subjects. Even for mathematics students, it's a great refuge from the excessive abstraction of most of the DG books out there.I used to turn up my nose at the Schaum's Outline series in the 1970s, but now I see that they are ideal for their purpose. Some people don't have a spare 3 years to read dozens of books and attend hundreds of lectures just to get a few basic formulas. There has to be some kind of literature for people who have a real life which they don't want to fritter away on philosophical minutiae. And I guess the Schaum's Outline series is ideal for that. This book is for the 99% who don't intend to do a PhD in mathematics. (And I refer to this book for my work too.)

I am approaching the subject from a mechanical engineering perspective- I have been through the pains of learning tensors, but mostly in a Cartesian sense. I used this book as a supplement in a course on Green Elasticity, and boy, were my eyes opened to "real" tensor analysis. I'm not sure if Green's book was that bad, or if this book was that great, but the fact is I was able to learn and understand so much more from this one. Generalized tensors is a difficult topic- there's a lot of seemingly arbitrary definitions and apocryphal notation. There just isn't a good book out there (to my knowledge), so the only way to learn it is by seeing and working a lot of examples by yourself. This is where this book excels- the "solved problems" sections in this book are better described as "simple examples and worked proofs," and I will be among the first in line to say that Professor Kay has done a very admirable job in explaining this topic. This isn't to say that some things he did are still a little muddy.I don't think I can call myself an expert yet, but I certainly have a much better grasp on the subject after reading this book and working some of the problems. I can certainly explain a lot better what is going on in the Cartesian tensor analysis that I am most familiar with. I don't know if this is recommendable to physicists, but if you are studying engineering mechanics and you want to know how deep the rabbit hole really goes, I would say this book is a fantastic place to start (though I would recommend keeping Geometrical Vectors close at hand.)