A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition by James G. Simmonds | (PDF) Free Download

Description

In this text which gradually develops the tools for formulating and manipulating the field equations of Continuum Mechanics, the mathematics of tensor analysis is introduced in four, well-separated stages, and the physical interpretation and application of vectors and tensors are stressed throughout. This new edition contains more exercises. In addition, the author has appended a section on Differential Geometry.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I can’t understand why some reviewers like this book so much. On the plus side, what is explained is explained clearly. For some struggling with tensors, this is no doubt helpful. But for me, a number of points were belabored unnecessarily (e.g. the first 15 pages on vectors) at the expense of developing more interesting material (notably topics related to relativity). Possibly the book deserves 3 stars but I have given it only 2 stars to counterbalance the highly positive views.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

⭐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.______________________________________________________________________Bottom line:Whether you are interested in classical mechanics or relativity, in my view, you would be far better served to read Kay’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

⭐(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.

⭐Simmonds has tried to create a primer for the beautiful world of tensor mathematics, and I suppose to that end he has met his goal. I do have to call him out on a lie though; in the first paragraph of the first chapter, Simmonds says he is writing the book in the context of continuum mechanics….this is very much false. The emphasis is on tensors as used by physicists studying general relativity, complete with contravariant and covariant forms of vectors and tensors and the unusual notation that comes along with it. Sometimes the explanations get a little muddy and overwhelming, and required a second or third read from me. However, the book still has a few high points for all users of tensors. Simmonds does a good job of keeping the discussion grounded in a physical and/or geometrical setting, something of which I’m sure engineers and mechanicians will appreciate. The introduction provides an alternate definition of a tensor than the usual Catch-22 “a tensor is something that transforms like a tensor,” which really helped me learn the subject matter. Each chapter has a good smattering of exercises which demonstrate the different concepts discussed in the text. Although a bit on the expensive side for about 100 pages, this is probably a good primer for

⭐. There’s a few typos along the way, but nothing major. If you want an introduction to general tensor analysis, this is probably the book for you.

⭐This is the clearest presentation of tensor analysis that I have ever read. The motivation of contravariant and covariant vectors becomes crystal clear.

⭐This book provides a first look at tensors. Its main virtue is that it draws a lot on what students already know about vectors and Cartesian spaces. That said, there is a lot of silliness in this book. The main one is that the author refers to superscript indices (aka contravariant) as “roof” indices, and subscripted indices (covariant) as “cellar” indices. Frankly, this is laughable. Is the book intended for third graders? Why not use accepted vocabulary? One does not need to go into “one-forms” and other more technical jargon, but use accepted vocabulary/jargon, it really isn’t all that hard.In Chapter 3 on Newtonian mechanics, the author uses the letter L, to indicate momentum. Why? Why not p or P, which is what every other book on the planet uses! The discussion of the Christoffel symbols is rather clunky, and the transformation law for the Christoffels on page 65, while true, is not at all useful. In fact, throughout the whole discussion of Newton’s law, recast into tensor form the equations are terribly dense, and when you arrive at the punchline, it is not at all obvious that you have or how you might make use of the result.Although the covariant derivative is discussed, its central role in GR, and its transformation properties are not discussed. Curvature is mentioned, but not really discussed. So, many of the topics that are important in either special relativity or general relativity are not adequately presented.All in all, the book is not really suitable for a physicist at any level. Many general relativity texts have a better intro. I would especially recommend a somewhat unknown text by Foster and Nightingale (

⭐especially Chapter 1. This latter text introduces the whole idea of tensors, first in a flat space from a geometric point of view, then in a curved space (surface of a sphere), again from a geometric viewpoint, and then generalizes to the 4-space of spacetime. There are other GR texts that also do a good job as well.

⭐It is a well done work on the brief side. But by me the mixing of index notation and vector forms, without an appropriate discussion, makes the reading difficult for not mathematicians in middle chapters. I suggest to use it for a second reading or successive reviewing on the subject.

Keywords

Free Download A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition in PDF format
A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition PDF Free Download
Download A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition 1993 PDF Free
A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition 1993 PDF Free Download
Download A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition PDF
Free Download Ebook A Brief on Tensor Analysis (Undergraduate Texts in Mathematics) 2nd Edition