- Series: Undergraduate Texts in Mathematics
- Hardcover: 114 pages
- Publisher: Springer; 2nd edition (July 31, 1997)
- Language: English
- ISBN-10: 038794088X
- ISBN-13: 978-0387940885
- Product Dimensions: 6.1 x 0.4 x 9.2 inches
- Shipping Weight: 9.6 ounces (View shipping rates and policies)
- Average Customer Review: 3.5 out of 5 stars See all reviews (9 customer reviews)
- Amazon Best Sellers Rank: #1,450,430 in Books (See Top 100 in Books)
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A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition
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Here's why I found the book wanting and in some ways, annoying.
(1) The coverage is really thin and some exposition relies on solutions to exercises but no solutions are provided.
In more detail: including the exercises the book has about 107 pages of which about 84 pages are exposition. So the exercises are roughly 21% of the book's content. A lot of the most useful and interesting material is left to exercises, e.g., derivative of a 2nd order tensor (Ex 1.24 p. 23), raising / lowering of indices (Ex 2. 9 (p. 39), notion of torsion (Ex 3.4 p.66), divergence of a tensor field (Ex 4.5 p. 98), Laplacian (Ex 4.10 p. 99), trace of 2nd order tensor (Ex. 4.15 p. 100), but there are no answers to any exercises. Worse yet, in places the exposition relies on solutions to exercises, e.g., Ex 2.10 p. 39 is used in Ex 4.17 p. 100 and the latter is a key element in the (interesting) discussion on p. 80 (4.24)! Unsolved exercises is one thing but reliance on unsolved exercises as an integral part of the mathematical development is really an all too common, lamentable practice.
(2) I doubt one could really learn anything much about Differential Geometry from the overly brief discussion on pp. 87-97. But this too is a significant amount of the book.
(3) The book is overpriced.
Personally I have found it more productive to study the far more economical book by Kay Schaum's Outline of Tensor Calculus (Schaum's) to learn the mechanics of the components-based approach necessary to understand physics books (and to work problems). In contrast to Simmonds, Kay has chapters on, e.g. The Metric Tensor, The Derivative of a Tensor, Riemannian Curvature, and even Tensors in Special Relativity. Besides covering more topics, he provides lots of examples and solved problems.
(4) The author introduces his own idiosyncratic terminology. This might be petty but it bugged me.
Specifically, he introduces his own terms for the standard terms "contravariant" and "covariant", namely, "roof" (= "contravariant") and "cellar" (= "covariant"). Because I was reading his book to understand some physics books, I kept having to translate back to "contravariant" and "covariant". Why not just remember "t is for top = contravariant", as I read somewhere, and "covariant" is the opposite? This area is already cluttered with different terms and causing further confusion with one's own terms is, to me, inexcusable.
(5) Pace the reviewers who regard this book as either oriented towards or useful for studying relativity (whether special or general), I found this book by and large without merit for understanding tensors in relativity.
Although it is true that the "alternative" formula for Christoffel symbols on pp 59-60 is applicable to general relativity, as the author states, there is no discussion in the book of indefinite metrics or four-vectors, etc., required for relativity, let alone any discussion of metrics in curved spacetimes (general relativity). General relativity is also mentioned in passing in ex. 3.12 on p. 68, and there's a section on the Metric Tensor (p. 89) and some discussion of geodesics (pp. 89-90) but the exposition uses the example of a "greased string stretched between P and Q". I can't quite see how any of this really prepares one for tensors in relativity per se.
Whether you are interested in classical mechanics or relativity, in my view, you would be far better served to read Kay's Schaum's Outline of Tensor Calculus (Schaum's). If you're interested in relativity, Kay is certainly the way to go, because it explains in some detail key topics such as indefinite metrics (CH 7.1), null curves (ch 7.3), and as mentioned above has entire chapters on Riemannian curvature (Ch 8) and Tensors in Special Relativity (Ch 12). Finally, if you'd like to understand the modern non-components view, i.e. tensors as multilinear functions on vector spaces, then you really need to study some standard book on differential geometry like John Lee's Introduction to Smooth Manifolds (for more suggestions on differential geometry books, cf. my reading list on that topic).
Well, usefulness is in the brain of the reader but unless you spot a great sale online, I'd get it from a library.
This edition seems geared towards applications in General Relativity. Though the chapter on Newton's Law shows how tensors can be used in material science deriving from classical physics, like with the use of the stress tensor.
The exercises in each chapter are fairly numerous. Most should not be too hard to the reader. Note that no answers are supplied in the text.