Tensor Analysis on Manifolds

By Samuel I. Goldberg and Richard L. Bishop

Book Overview

"This is a first-rate book and deserves to be widely read." -- American Mathematical Monthly
Despite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject. The authors have treated tensor analysis as a continuation of advanced calculus, striking just the right balance between the formal and abstract approaches to the subject.
The material proceeds from the general to the special. An introductory chapter establishes notation and explains various topics in set theory and topology. Chapters 1 and 2 develop tensor analysis in its function-theoretical and algebraic aspects, respectively. The next two chapters take up vector analysis on manifolds and integration theory. In the last two chapters (5 and 6) several important special structures are studied, those in Chapter 6 illustrating how the previous material can be adapted to clarify the ideas of classical mechanics. The text as a whole offers numerous examples and problems.
A student with a background of advanced calculus and elementary differential equation could readily undertake the study of this book. The more mature the reader is in terms of other mathematical knowledge and experience, the more he will learn from this presentation.

• Format: Paperback
• Language: English
• ISBN: 0486640396
• ISBN13: 9780486640396
• Release Date: December 1980
• Publisher: Dover Publications
• Length: 288 Pages
• Weight: 0.71 lbs.
• Dimensions: 0.6" x 5.6" x 8.3"

Customer Reviews

A bit difficult for the non-professional but overall a fascinating book

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. :-)

"Did you say relativity?" Find all the prerequisites here

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.

Well-written text

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 ofconnection (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.

good introductory text

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.

Terse, clear, modern introduction to tensors and forms

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.