Something good to say about geometry, for once
I just realized that Frankel’s book, which has been sitting untouched on my shelf for a year or so, is the perfect complement to Marsden’s book. First, it is approachable– here’s a book that’s actually aimed at engineers and working scientists– and second, it is almost as comprehensive as Marsden’s book, in terms of concepts introduced, but less mathematically rigid. The feeling I’m getting is that I can read this to develop an intuition for dealing with the concepts (Lie derivatives, etc.) that Marsden’s book covers in too much detail to be digestible to first timers, then jump into Marsden’s text to see rigorous developments if I care to. Which I probably won’t, at this point.
But, the most important point, and the one putting a smile on my face as I type these words, is the fact that working in coordinates is not as much of an afterthought in Frankel’s book as it seems to be in Marsden’s. So, e.g. I can learn about tensors in a natural manner, where the coordinate manipulations are developed naturally with the more intrinsic operations. Considering that the geometry final is going to be almost exclusively on manipulation of coordinate formulas, this is a big plus for me.
Possibly relevant posts:
- Definitions (5/9/2005)
- Change of coordinates on manifolds (9/20/2006)
- the insanity that is CDS202 (1/26/2008)
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.
Comment by ObsessiveMathsFreak — 2/27/2008 @ 9:02 am
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.
Comment by Alex — 2/29/2008 @ 11:57 am