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A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition Hardcover – January 1, 1999

ISBN-13: 978-0914098706 ISBN-10: 0914098705 Edition: 3rd

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Product Details

  • Hardcover
  • Publisher: Publish or Perish; 3rd edition (January 1, 1999)
  • Language: English
  • ISBN-10: 0914098705
  • ISBN-13: 978-0914098706
  • Product Dimensions: 9.2 x 6.7 x 1.3 inches
  • Shipping Weight: 2.3 pounds (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #161,902 in Books (See Top 100 in Books)

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110 of 115 people found the following review helpful By Paul Thurston on September 6, 2005
Format: Hardcover Verified Purchase
This book is the first volume of the 3rd edition in a five volume series on differential geometry. The emphasis on this first volume is the study of differential forms and de Rham Cohomology Theory. Spivak also considers two 'bonus' topics: integral manifolds & foliations and Lie groups.

You'll need some prerequisites to get started. For the differential topology material (including Sard's Theorem and Whitney's 2n+1 Embedding Theorem), I recommend Hirsch's Differential Topology. For results on determinants and symmetric groups, I use Hungerford's Algebra, now in its 12th printing. For the general topology material (Hausdorff spaces, Urysohn metrization, etc.), I recommend Munkres' Topology (2nd Edition).

Spivak begins this volume with a review of topological manifolds in Chapter 1. The author provides the basic definitions and gives lots of examples of surfaces and other manifolds. The discussion of manifolds and surfaces continues in the Chapter 1 Exercises. (The author routinely used the exercise set to continue the thread of discussion.) Quick mention of the surface classification theorem is made, although for the proof of this, you'll need to look in Hirsch or Munkres. The reader gets to have fun gluing topological handles onto and cutting disks out of the 2-sphere.

Chapter 2 reviews some of the basic concepts from differential topology, including the fundamental Whitney Embedding Theorem and Sard Critical Point Theorem.
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52 of 60 people found the following review helpful By Fadi E. on September 27, 2007
Format: Hardcover
Spivak's text gets a lot of good reviews, and it is a fine text. In fact, it's one of the best I've ever seen. Read a few other books on the subject, and you'll agree that this is a massive improvement on them. So why only 3 stars? Because there's a much better text on the subject: John Lee's "An Introduction to Smooth Manifolds". This book outshines Spivak's in so many ways. Sure, Spivak is great at motivating major developments in the theory (for instance, he really helps you understand why we need to define a tangent space and why it is the way it is), but he fails pretty bad when it comes to developing some actual theory.

Reading Spivak's text is like taking a stroll, a fresh break from the usual mathematics textbook style. But you also hit a bunch of brick walls on this stroll. It'll be a great discussion, and then you'll come to a theorem. You'll have no idea what its for (some of the time) and you'll struggle to work through its proof (most of the time). Furthermore, the organization is... well, there is no organization! As a result, Spivak can seem to droll on. Lee isn't as good at giving the overall big picture as well as Spivak, but he does everything else exceptionally. Leave Spivak for bed time reading, but do your real studying out of Lee.
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66 of 79 people found the following review helpful By Steven T. Smith on July 9, 2003
Format: Hardcover
Michael Spivak begins these five volumes stating his modest aim to write the "Great American Differential Geometry book." He surely has. Instead of listing the numerous subjects Spivak treats clearly and beautifully in these volumes, I'd like to call out the delightful travelogue style in which they are written, using history, anecdotes, and opinion to explain, illuminate, and, when possible, motivate the gleaming modern edifice. Spivak's opinions are sprinkled lightly here and there like easter eggs. How could you not love a math book that uses the subtitle "The Debauch of Indices," or dismisses Eric Temple Bell's history as "supercilious remarks of questionable taste"? Also, don't miss the annotated bibliography in volume 5. The fact that legions of professionals refer to these books in their original *typewritten* format [1st & 2d editions] is a further testament to their quality. The third edition is typeset using TeX and, though beautiful, still manages to retain a little of the quirky typewritten appearance. One quibble: I was disappointed to see that this edition did not use Richard Bassein's bizarre artwork [think 70s psychedelic] for the covers; I admit that this stuff weirded me out originally, but have grown to love it -- where else could I see fuzzy trolls in crowns made from Enneper's minimal surface?
Let Spivak take you "All the Way With Gauss-Bonnet."
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13 of 20 people found the following review helpful By C. Katouli on February 22, 2006
Format: Hardcover Verified Purchase
If you want a book that is rich with examples then this is it. The proofs are, for the most part, clear and concise, thus a person who is learning the material without the aid of an instructor can follow the logic. However, the author could have spent some more time developing topological ideas (thought he does have an appendix section that does a fair job of it) within the flow of the first chapter. I personally find appendices to be too distracting and tend to slow down the flow of the material in a particular chapter. Other than that, this is a great book if you want to learn differential geometry and the theory of smooth manifolds.
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