Comments on: Something good to say about geometry, for once http://www.tangentspace.net/cz/archives/2008/02/something-good-to-say-about-geometry-for-once/ somewhere near the beginning. Fri, 21 Nov 2008 21:32:52 +0000 http://wordpress.org/?v=2.5 By: Alex http://www.tangentspace.net/cz/archives/2008/02/something-good-to-say-about-geometry-for-once/#comment-270228 Alex Fri, 29 Feb 2008 18:57:21 +0000 http://www.tangentspace.net/cz/archives/2008/02/something-good-to-say-about-geometry-for-once/#comment-270228 I have had the same thought about the impracticality of starting off with manifolds. Sharpe's differential geometry book is good in that respect-- he develops the ideas like tangent spaces, derivatives, etc. using the level set/gradient definition. Then at the end, I think he goes into smooth manifolds. I have had the same thought about the impracticality of starting off with manifolds. Sharpe’s differential geometry book is good in that respect– he develops the ideas like tangent spaces, derivatives, etc. using the level set/gradient definition. Then at the end, I think he goes into smooth manifolds.

]]> By: ObsessiveMathsFreak http://www.tangentspace.net/cz/archives/2008/02/something-good-to-say-about-geometry-for-once/#comment-268244 ObsessiveMathsFreak Wed, 27 Feb 2008 16:02:10 +0000 http://www.tangentspace.net/cz/archives/2008/02/something-good-to-say-about-geometry-for-once/#comment-268244 Marsen put me to sleep before the end of the first chapter. It's too topological. After that bad start, I had little time for Frankel's discussion of manifolds at the beginning of his text. The first chapter really is terrible. Frankly, I think it's entirely wrong for differential geometry textbooks to begin with discussions of manifolds. There's really very little point in introducing them to newcomers when regular surfaces are perfectly sufficient at that stage. The theory and definition of manifolds is entirely indigestible, and totally abstract, for any beginner. As a concrete example of their abstractness having read quite a few differential geometry textbooks, I have never even once seen a proper, explicit and machine readable coordinate atlas for the sphere, complete with derivative functions. Never. Authors speak of such things abstractly, but never use them. In other words there really is no point discussing them until later. Considering your appraisal of later chapters, I might pick up Frankel again when I am less busy. I think I'll start at chapter 2. Marsen put me to sleep before the end of the first chapter. It’s too topological. After that bad start, I had little time for Frankel’s discussion of manifolds at the beginning of his text. The first chapter really is terrible.

Frankly, I think it’s entirely wrong for differential geometry textbooks to begin with discussions of manifolds. There’s really very little point in introducing them to newcomers when regular surfaces are perfectly sufficient at that stage. The theory and definition of manifolds is entirely indigestible, and totally abstract, for any beginner.

As a concrete example of their abstractness having read quite a few differential geometry textbooks, I have never even once seen a proper, explicit and machine readable coordinate atlas for the sphere, complete with derivative functions. Never. Authors speak of such things abstractly, but never use them. In other words there really is no point discussing them until later.

Considering your appraisal of later chapters, I might pick up Frankel again when I am less busy. I think I’ll start at chapter 2.

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