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Curvature in Mathematics and Physics (Dover Books on Mathematics) Paperback – August 22, 2012
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Top Customer Reviews
I'm the CTO of Classpros dot com and design math visualizations for Engineering students and this is NOT an undergrad text, unless you're an EXCEPTIONAL undergrad with a LOT of physics and calc already behind you, as in a senior at a tech college. This is not at all to knock the book, it's to save purchasers from disappointment when they see the author jumping into Relativity from the viewpoint of Cartan exterior calculus and tensors.
That said, this is an EXTRAORDINARY text because differential geometry has become so specialized that few grad students except in limited areas of physics/applied math get to go there. The "expansion" of the field with game programming and sims is a new revolution that is "bringing back" fields as dusty as quaternions and spherical trig (see our article on Wiki on the Lenart Sphere, for example)-- and creating exciting new interest in differential forms and geometry.
After reviewing and using half a dozen (rare and old) books on differential geometry, this is the ONE book you must have if you are serious about the field, both for breadth and depth and especially currency. Both the most recent applications and the older physics are covered flawlessly. Also, if this were a Springer text, it would be over $100 at this quality and rarity-- what a bargain from Dover!
Several reviewers have noted this text as worth the price even though it is a "classic." Ahem.Read more ›
Here it is:
1. Gauss's Theorem Egregium
2. Rules of Calculus
3. Connections on the Tangent Bundle
4. Levi-Civita's Theorem
5. Bi-invariant Metrics on a Lie Group
6. Cartan Calculations
7. Gauss's Lemma
8. Variational Formulas
9. The Hopf-Rinow Theorem
10. Curvature, Distance and Volume
11.Review of Special Relativity
12. The Star Operator and Electromagnetism
13. Preliminaries to the Einstein Equation
14. Die Grundlagen der Physik
15. The Frobenius Theorem
16. Connections on Principal Bundles
17. Reduction of Principal Bundles
19. Semi-Riemannian Submersions
Far more important for me was brushing-up on (and then tightening-up on) both linear algebra and Group Theory, -- at least up through Lie Groups. Breathing space for this was more or less allowed in the problems section at the end of chapter 2. Without this warm-up session, trying to apply Riemann geometry to a Euclidian space in chapter 5 would have amounted to little more than a "distant abstraction."
The challenge throughout the book was making the connection in ones mind of the dizzying array of curvatures associated with hyper-surfaces possible to embed in Euclidian space. I finally figured out that doing this through the "Calculus of Manifolds" is what this book was all about, and thus represented the true value of the book?
In any case, whether true or not, I was delighted to see both the "Lie Derivation of Vector Fields," and the "Schwarzschild Solution" worked through. Previously I had never seen a proper analysis of the former, and the latter only as "settled formula" -- without ever having seen its derivation.
So far, I have completed the book up through chapter 12, more enlightened than frightened.
When the masters of a subject (such as Shlomo Sternberg is here) presents his material, there is the natural expectation that the reader is going to have to stretch a bit to keep up.Read more ›
Most Recent Customer Reviews
The book covers very interesting topics in differential geometry with applications to theorytical physics, in particular the thoery of relativity. Read morePublished 10 months ago by Samir
The advances of the last 200 years in Mathematics and Physics are all based on the realization that there are critical defining concepts that give meaning to the statements. Read morePublished 19 months ago by kikeo58
This book gives a lot of very good details in a lot of areas but I particularly liked the coverage of torsion For the price you cannot beat it.Published on December 18, 2013 by Joseph C. Walker
Much too advanced mathematically for general readership. Illustrations are well done, giving a possibly
intuitive grasp of the concepts presented.