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Curvature in Mathematics and Physics Format Kindle
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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. Dover DOES reprise many worthy volumes, but this $13 gem is NEW IN 2012-- yes, by a professor that's taught it for half a century, but this is NOT one of Dover's well known reprints from 1956, it's new and original! In fact, the author (humbly) compares it to the "gold standard" of O'Neill (from 1982), which IS $113 even on Amazon (Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics)). If you're a prof considering a pre-relativity course, please do your students a favor and consider this fine text! Dover is trying to set a trend by offering very high quality texts for prices us un-rich folks can afford! With all due respect to O'Neill, at 10% the cost, nearly the same pages, and more intuitive notation-- why not? Give your class a ramen free weekend!
Library Picks reviews titles exclusively for the benefit of Amazon shoppers and has nothing to do with Amazon, the authors or publishers, and we always buy the books we review. Since another reviewer published the contents, you've probably got what you need to decide. If you've had to figure out what is integratable or not based on whether you can find linear differentiable forms using linear algebra and other tools, you're ready for this fine text. Otherwise more advanced calc and linear algebra are musts first. Unless other reviewers are channeling Albert, I'm not sure why they're trying to tell you this is "elementary?" Guess the concept is relative, pun intended.
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. And although this book is slightly over the head of my dwindling mathematical skills, the challenge of understanding the mathematics in Einstein's formulation of General Relativity alone forced me to persevere. I paid $250 for a book that consists of photo copies of his original mathematical derivations and intend to fully understanding it before I die. Thus this book was an unexpected part of that life-project. There was the added impetus too that when I browsed the book in Barnes and Noble, I was at least minimally familiar with most of the topics introduced as its content. Although I will admit that some of it was heavy going.
That said, I am not sad I embarked on this challenge, because up through the chapter on Geodesics, although the names had changed from what I recall in graduate school Topology, the methodology and methods of proofs were exactly the same. Even the tensor analysis, which simply involved generalized matrix mappings, presented no problem. Yet, there always (whenever matrix manipulations are involved), the many bookkeeping problems. The myriad problems of keeping track of all the indices as one ascends levels of abstraction. My rule of thumb of alway reading the formulas in as many alternative ways as possible before settling on a single interpretation, seems to have served me well.
AT $19.95, this is probably the best buy I have gotten on a book in the last decade. Five stars.
The book uses the modern definition of "differential manifold" throughout, but I can't find it defined anywhere in the book. The grossly inadequate index contains only 17 items starting with "m" , and these do not include "manifold" ! The closest to a definition seems to be a definition of "parametrized surface" in Chapter 1.
I doubt that anyone without a previous acquaintance with differential forms will get much out of this book. Technically, the definitions are given in Chapter 2, but in only a few pages in a very abstract way. Some of that treatment seemed to me downright perverse. For example, the concept of "Lie derivative" of a vector field has a simple geometrical interpretation given in almost all texts. From this follows the related concept of Lie derivative of a differential form field, whose geometrical motivation depends on the previous concept of Lie derivative of a vector field. But Sternberg introduces these concepts "backward" starting with a geometrically unmotivated algebraic definition of Lie derivative of a form field (which relies on the nontrivial Weil formula) and from that produces an algebraic definition of Lie derivative of a vector field. I would be surprised if anyone unfamiliar with the general concept of "Lie derivative" will come away from this discussion with an adequate understanding.
That said, there is a lot of value in this book for those already familiar with differential geometry. I enjoyed browsing through it. It includes many pictures of prominent differential geometers, with some biographical sketches. Most of the mathematics is presented at an appropriate level of rigor, perhaps not quite as high as mathematical research, but much higher than typical physics literature. Though the exposition is more informal than that of most mathematics texts, it is usually no less rigorous.
Apart from applications to general relativity, most of the physical applications were presented so sketchily that they were hard for me to follow in detail. The "Curvature in ... Physics" part of the title should be taken with a grain of salt by readers without extensive experience in the relevant fields of physics.
I cannot recommend this book as a textbook, but I have enjoyed browsing through it and am happy to have it in my library. I think that for its price, it is a bargain.
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