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Tensor Analysis on Manifolds | 
 enlarge | Authors: Richard L. Bishop, Samuel I. Goldberg
Publisher: Dover Publications
Category: Book
List Price: $12.95
Buy New: $7.93
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Rating:  13 reviews
Sales Rank: 183177
Media: Paperback
Pages: 288
Number Of Items: 1
Shipping Weight (lbs): 0.7
Dimensions (in): 8.4 x 5.9 x 0.6
ISBN: 0486640396
Dewey Decimal Number: 515.63
EAN: 9780486640396
ASIN: 0486640396
Publication Date: December 1, 1980
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Product Description
Proceeds from general to special, including chapters on vector analysis on manifolds and integration theory.
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| Customer Reviews: Read 8 more reviews...
 A solid text on manifolds October 1, 2008
R. Bagula (Lakeside, Ca United States)
I have some reservations:
1) no notation/ symbols page
2) uses the Klein Map but doesn't give the name
3) no clear affine and projective manifold classification
But in contrast it has good handling of diffeomorphisms and integrations
on manifolds.
I bought it to the Weeks space manifold, M003[3,-1], and the polyhedron forms that Weeks uses. It isn't really
of use in that more modern area of manifold theory either.
The Thurston space manifold, M003[2,-3], isn't covered.
The book is a good introductory text that I wish I would have had 40 years ago!
 Tough for self-study June 2, 2008
Art K. (San Jose, CA United States)
1 out of 1 found this review helpful
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.
 Fantastic May 16, 2008
Kevin
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.
A bit difficult for the non-professional but overall a fascinating book May 8, 2008
magellan (Santa Clara, CA)
3 out of 3 found this review helpful
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. :-)
Great book, horrid notation May 5, 2008
Kevin A. Brown
3 out of 3 found this review helpful
The amount of material this book manages to cover is phenomenal, and while the explanations are very clear, the summation convention notation is horrible. The idea is that things are clearer if we assume that we are summing whenever we see two of the same indexes in an expression (which sometimes has 5 or more), which leads to many headaches when trying to sort out when we are summing and when we are not, because it's not always very clear, and sometimes very important to a particular proof or concept. Maybe I learned so much from this book because I was forced to rewrite all of the notation, guessing whether sums were needed or not, and finding out when the sums made sense.
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