Customer Reviews

Schaum's Outline of Tensor Calculus (Schaum's)

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This Study Guide functions properly. If right now, you are reading the title "Tensor Calculus" and wish you understood it someday. But have absolutely no idea what a tesnor of rank zero is. GET THIS BOOK

My main goal was to understand General Relativity. But as you know, the mathematics of General Relativity is nothing but Tensor Calculus. I was particularly intrigued by the mysteries of the Riemann Curvature Tensor. The key to General Relativity. As soon as I purchased this book, I started studying Chapter 8, the "Riemannian Curvature" not knowing anything about the previous chapters. Hopeless I eventually turned to chapter 1 and gradually climed up the ladder. Then came my Golden Times in Tensor Calculus. I cracked the mysteries of the Riemann Curvature Tensor and at last I turned to General Relativity. I'm currently studying black holes thanks to this book.

This is probably the clearest ontoduction to tensor analysis that is currently on the market. It makes a quite difficult and messy subject seem pretty straightforward. It's best to know your vector calc in and out before attempting this book, but it's a godsend compared to some of the other texts out on the market today. A great guide for engineering and physics students and the price can't be beat.

Tensors are a booger to learn, even with a great guide like this. It's just hard. But this is the best guide available. More completely worked out problems would be nice. A very very solid grounding in vector analysis and linear algebra is required before tackling this book, however.

This is not a book to learn tensor calculus from. It is an outline only, no greath depth or insight is presented. This book works perfectly as a supplement to a course in tensor calculus, or as a quick reference for the various techniques and concepts involved, provided one is already somewhat familiar with the material. It would be possible to learn the basics of tensor calculus from this book with some effort, and reflection on the implications of the concepts dealt with, however as a complete course in the subject it is insufficient, and I believe intentionally so.
The more modern aspects of tensor analysis on manifolds are largely ignored in this treatment, but also intentionally so, an approach which I found useful practically.
The book does not aim to be an all-inclusive course in the applications of tensor concepts to all areas of mathmatics, but rather a quick-reference guide supplementing more complete treatments, and as such, is largely successful.

So far, I've covered the first 5 chapters. Very easy to understand, and plenty of example problems. What more could you ask for?

I have been studying a textbook in general relativity. A lot of the material that they gloss over is detailed in an understandable way in this volume.

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 10 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.

Another book I like is Tensor Calculus, Second Edition. Recently I noticed a new contender, which looks especially well-suited for relativity enthusiasts, so I'd check this one out too: Tensor Analysis With Applications.


Lastly, here are two books on tensors that I found to be not helpful 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.

If it is desired to get rid of much of the reluctance in tensor calculus and fight powerfully with Einstein index summation and relief the pains with Christoffel symbols and Riemann Curvature Tensor, essential for General Relativity understanding then, I think this book, with solved problems and many others for practicing, is a good guide to make General Relativity and Cosmology more pleasant. I am a self-learner of Cosmology and of these kind of mathematical topics, I used many books from the more theoretical to simpler ones and think that this and the Sokolnikoff's (Tensor Analysis and Its Applications) are a good couple of references to understand this branch of tools for Physics and making me more happy dominating the practice with these objects. Later on, in case of more interest in going deeply in the axiomatic bases of tensors (possibly by using Gravitation of Misner, Thorne and Wheeler), at least one can already have the calculus foundations to go on. I know many other excellent books (and I would not say that this is the best), but to me this is the one I found better to condensate the most of all my doubts in the subject and.... go on.

I've many books about tensors and this one is the best one to start learning such a difficult subject. It does not omit the things that are assumed you must know. It explains everything, even the simplest things in a easy way. However, you should know vector analysis and multidimensional calculus in order to understand the complex things in the last chapters.

Before I had this book I knew nothing about tensors. Other books were old and useless. This book was very easy to understand and explained everything I needed to know. Now I work with tensors in General Relativity all the time!