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Differential Geometry of Curves and Surfaces (1976)
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Manfredo DoCarmo
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Pearson
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1 edition
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503 pages
ISBN
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8945045090
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960 Point
This volume covers local as well as global differential geometry of curves and surfaces.
1.Cuves
2.Regular surfaces
3.The geometry of the gauss map
4.The intrinsic geometry of surfaces
5.Global differntial geometry
This volume covers local as well as global differential geometry of curves and surfaces.
"
This book is rather expensive, but when compared to the other books available, it is not a waste of your money. It has plenty of exercises, many of them with answer or hints in the back of the book, and its exposition is broad, very clear and concise.
It is hard to tell being a math student, but I think anyone with a solid knowledge in multi-variable calculus (Apostol's book would be perfect) or, better yet, who has taken multi-variable analysis course would find this book accessible. One of the advantages of this book is that it is self-contained, so even though it uses, for example, the inverse function theorem (which is something unavoidable for a DG book), it has an appendix on differentiability and continuity which covers this.
The exercises range from easy to very hard, but because of the exposition and of the way the exercises are stated (the tougher ones are many times itemized so that they drive you to the answer) it is rare to find a problem that the reader will not be able to solve upon a little thinking.
The greatest advantage of this book is its clear and well-written exposition. It has very few errors, mostly typographical. It covers a lot of topics and its notation is extremely simple and suggestive, which, believe me, is of great help in a DG book. In short, if you want or have to learn differential geometry, save your time and get this book. As another reader very intelligently put it, there is a reason why this is a classic. "
"
Before talking about the book itself, let me tell you that I am a mathematician, and when I took a differential geometry course and used do Carmo's book, I already knew I wanted to be a mathematician. So, is this a book for mathematicians? Well, yes, but not exclusively. It is certainly written from a mathematician's point-of-view, and it assumes some maturity on the part of the reader. For instance, the exercises contain very little in the way of drill, and are used to enhance the theory (as pointed out by another reviewer). It seems to me that the author believes that mature readers can provide their own `drill' exercises. So, you won't find many exercises asking for you to find pricipal curvatures for this or that surface, and that other as well; exercises in this book have a theoretical flavor to them. This, of course, makes for some hard exercises, and I do remember spending a lot of time over them, often working together with other students taking the same course. The upside is that we learned the material, and thoroughly. Also, the author provided plenty, plenty of examples. The figures are very well drawn and really allow you to see what is going on - even though these days, with powerful computer packages like Maple, Mathematica, Matlab, and others, any student can provide his/her own pictures. But just because now we can use computers, I wouldn't say the text shows signs of age. It is jus as clear now in its exposition of topics and concepts as it was many years ago. So, even though there are many good alternatives in the market, if I had to teach a course now on this subject, or even better, if I were a student now taking this subject, I would certainly have this book at the top of my list of possible textbooks. "
"
An earlier review said this book has few errors, and even then only typographical ones. Are we talking about the same book? The text is pockmarked - nay, cratered - with scads of dire gaffes. The skeptical empiricist should go to Google and enter these keywords: bjorn carmo errata. The first hit will be a link to a 7-page pdf file a U.C. Berkeley professor and his students created a few years ago which compiles errata they turned up. Seven full pages, and they only covered a third of the text! A sample item in the list: "p. 97, definition of domain: It is not clear whether the boundary is the boundary as a subset of R^3 or the boundary as a subset of S. Either way, we run into trouble..." The Heine-Borel theorem on page 124 is so botched up it's beyond repair, and even the basic definition of what it means for a function to be continuous on a set is faulty (p. 123).
The author claims a student should be able to hack the material with "only the most basic concepts" from linear algebra and multivariable calculus. Largely but not entirely true. For example, you better be up to speed on linear mappings defined by NON-square matrices - something no undergraduate-level linear algebra book in my library discusses (though I only possess a handful). Many of those tidy little results for linear operators from R^n into R^n you might know from Linear Algebra 301 become worthless when one of those n's becomes an m. I don't really fault the author for this, but anyone thinking about acquiring this text should know it is not by any stretch "self-contained" as one previous reviewer stated.
The biggest irritant with this text is the constant abuse of notation. When you're just starting out trying to learn this stuff, it most emphatically does NOT help when the author keeps butchering or truncating the notation in the interests of "brevity". For example, entire derivations are often carried out using only the names of functions and not their arguments. Maybe I have a screw loose, but sometimes I find it really helps knowing that f is really f(x) and g is really g(y). And then in a single section the same symbol, N, is used to denote three different functions. Okay: this N really means N composed with alpha, and that N is N composed with the parametrization x, and this other N is really N all right, but when you stick q into it you really mean x inverse of q because N's domain is a plane, not three-dimensional space...yeah...oh wait...
Over the course of a semester I wasted uncounted hours unraveling DoCarmo's infuriating and uncalled-for notational "short-cuts".
So why do I give this text 3 stars? Well, in using it I still managed to master the core concepts quite nicely, so it had to be doing something right. Yes, the exercises are challenging, but I was able to crack most of the ones I attempted. Maybe a problem would take me 8 hours to do, but I could do it. I would not say this book is "dated", as one reviewer put it. In fact, I found most of the minimalist, non-computer generated figures to be refreshing and adequate. Anyone seeking cookbook methods for computing things will be sorely disappointed: this text is written largely for students of mathematics, and I have no qualms with that (being a math student myself).
Truly, if this text were given a buff and polish, it could become 5-star material (in my opinion anyway). Alas, that's probably not in the cards. "
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