Customer Reviews

Tensor Analysis on Manifolds (Dover Books on Mathematics)

5 star:		(8)
4 star:		(6)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

I will briefly list the pros and cons of this book.
The pros are (a.) its price, (b.) the amount of material it manages to cover, (c.) it is quite complete - everything is formulated and proven within the text rigorously, and it covers a lot of ground (manifolds, tensors, differentiation and integration on manifolds, connections and Riemmanian manifolds) (d.) it does not require much background - nothing more than point-set topology and calculus. It even develops all the linear algebra it needs in a single chapter - quite admirable. (e.) the exercises are nice and instructive. (f.) It makes a good reference and supplement. (g.) It has a special chapter on Riemannian manifolds - quite good for relativity courses.
Now for the cons. (a.) the notation is a bit outdated. (b.) it does not treat infinite dimensional or complex manifolds. (c.) It sometimes leaves certain results for the reader to verify, which might annoy readers who simply want to get to a certain result as quick as possible. (d.) It is a bit dry. (e.) It lacks in concrete examples - that is not to say it doesn't have any examples, just that more would be much better, (f.) and this is chiefly aimed at physicits - it does not really focus on calculating things, which is what physics is all about, at the end.
Having said that, I honestly say that one can learn all about basic differential geometry from this book. I don't think seeing manifolds in R^n is a basic prerequisite for studying abstract diff. geometry. This book would be a good place to start - despite its age it manages to remain very relevant today. Finally, the reader is assured that the authors won't pull off any "dirty tricks" (since this is basically a mathematical book) - it's very important for the reader to be able to trust the book he's reading. And the price is fantastic!

The best introductory book on its subject. Being a physicist, not a mathematician, I particularly appreciated its self-contained and down-to-earth, though fully rigorous, style. The very good chapter on integration of forms shows mastery of the authors both in the topic and in the technique of exposition. Terse, yet very clear: a rare combination that reminds one of the best books by Halmos.

This is a great book for an introduction to differential geometry. The only real prerequisite is calculus and some topology, making this book accessible to undergraduate students interested in Mathematics or Physics. The book covers a wide variety of topics and there are plenty of examples and exercises.
I guess the two reasons why I don't give this book five stars are (1)the notation in not entirely modern and (2) I have not managed to effectively use it as a primary textbook but as a supplument to a textbook.
It is certainly a great value for the price.

This books is the perfect introduction to modern differential geometry, especially for people with a specific purpose in mind such as the study of relativity or analytical mechanics. This book is a very straight forward read. But that dosent mean it compromises on quality on the depth of the material presented. The exercises are great, as they illustrate the concepts just learned very nicely. One section leads very nicely to the other. As for the topology needed to study differentiable manifolds, it is developed in the beginnning, though its not the best "quick untro to topology" Ive seen. Of course you can skip some of the sections such as Paracompactness. The only consequence is that you might not be able to follow some of the proofs later on. The only other complaint is that in the few exercises on special relativity, they use the old "ict" coordinate system. Try to remember that this sysytem is frowned upon these days. But all in all an excellent read. And especially for the price you can buy this at.

This is a terse treatment of differential geometry. It is perhaps too sophisticated to serve as an introduction to modern differential geometry. The beginner probably needs to see examples of two dimensional surfaces embedded in Euclidean 3-space and to do calculations with reference to such surfaces. For example,the use of coordinate patches to cover the 2-sphere. And then seeing how the change of coordinates in overlapping patches affects geometric objects such as vectors, 1-forms, and the metric tensor. This provides some grounding for the abstract treatment of manifolds and the tensors defined on them. Also a leisurely introduction to the geometry of curved surfaces, either classically, using the first and second fundamental forms, or the modern way, using the shape operator (which is equivalent). This motivates the more abstract treatment of connections, which become necessary when there is no underlying space to embed the surface in (Euclidean 3-space provides a notion of

connection (i.e. covariant derivative) that is geometrically clear; we have to axiomatize this notion when there's no natural space to embed in).

Though the book may not be suitable as a first text, it can be used in conjunction with a more elementary text. Alternatively, it could be used for a graduate course. Though there are now a plethora of other good treatments around, this book remains one of the classics,and furthermore its price makes it particularly appealing.

Of all of these dover books, this is the best one to learn differential geometry from (you can safely ignore just about any other dover text with the word "tensor" in the title). The authors keep the exposition very clear with good examples.

This is the unofficial "standard reference" on the subject. Lots of more advanced books cite it, and it certainly covers a lot of ground. The problem is it is a bit too terse. I say that as a math type. Were I a physics type I'd want a more visual approach. If you are learning tensor geometry though this book is worth having as a reference, and the price is right.

I came to this book with the minimum background--calculus and advanced calculus, differential equations, and some linear algebra, and found it a bit tough going, but still enjoyable. In fact, for me, not being a mathematician but a math hobbyist, really, whose education is mostly in biology and art history, I found it pretty difficult but also quite fascinating and even mind-blowing. I only had the vaguest ideas about tensors, fields, and manifolds before this, although I knew that the theory of manifolds underlies differential geometry and Einstein's famous General Relativity theory.

I understand that the notation in this book is considered old-fashioned and may contribute to the difficulty of reading it. Not having had anything different I don't know if it was harder for me or not, but overall I didn't find the notation too bad. The authors make the interesting point in the introduction that notational developments have occupied much of the work in manifolds, which I found funny. This implies that you can be good at math notation but not that good at the math. So maybe there's hope for me yet. :-)

That issue aside, I found this a very complete and well presented discussion on the subject. Some of it seemed pretty abstract and even counter-intuitive; for example, the concept of distance between two points isn't necessary to have a manifold, and yet having a coordinate neighborhood, or a manifold consisting of differentiable functions is, or other similar properties. It is a little strange to consider that one can perform differentiation on a manifold without the concept of spatial distance, when to my mind taking delta y over delta x at the limit is just shrinking the distance down to nothing in order to obtain the derivative of a function, not to mention that this seems problematic given the requirement of either uniform or non-uniform convergence. How do you know the function converges without some concept of distance? If you're better at this stuff than I am perhaps you could leave me a brief comment if I'm getting something wrong here.

But I still learned a lot, and much of it is pretty amazing and even mind-blowing stuff. People wouldn't need psychedelics if they knew enough to be learning about tensors, manifolds, and topology. They could blow their minds just on this stuff. :-)

So go out and get yourself a book on tensor manifolds and blow your mind the natural way. Higher mathematics is just awesome stuff even if I'm not quite smart enough to really understand it, but I can at least appreciate it, and I probably got a lot further with it than most biology and art history majors. :-)

I have been using this book to study differential geometry for many years - a little bit at a time. This book is a fairly complete introduction to the subject. However, it does a poor job motivating and explaining the subject. I found it necessary to supplement with several other texts to really get a good grasp on the material in the book. A number of times, I have picked up something in another book and have gone back to this book and realized that I had not "gotten it" the first time through. If the book had more examples and concrete calculations it would go a long way to clarifying the material.

I would recommend getting a book like Guggenheimer's Differential Geometry and reading it first. This book then does a good job of generalizing the ideas to many dimensions.

Great book. Very clear explanation of what manifolds are and how to use them. Also, the prerequisites are minimum. You'll get a lot of mileage out of this book if you have a semester or so of advanced calculus and some topology and linear algebra under your belt. I wish there was there was a chapter on bundles because I would love to read what these guys have to say about them. Very readable. Definitely a good buy.