- Series: Dover Books on Mathematics
- Paperback: 240 pages
- Publisher: Dover Publications; Second Edition edition (April 1, 1988)
- Language: English
- ISBN-10: 0486656098
- ISBN-13: 978-0486656090
- Product Dimensions: 5.4 x 0.5 x 8.4 inches
- Shipping Weight: 9.1 ounces (View shipping rates and policies)
- Average Customer Review: 4.7 out of 5 stars See all reviews (7 customer reviews)
- Amazon Best Sellers Rank: #569,910 in Books (See Top 100 in Books)
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Lectures on Classical Differential Geometry: Second Edition (Dover Books on Mathematics) Second Edition Edition
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First of all, this book is very readable, being that it requires no more than 2 years of calculus (with analytic geometry and vector analysis) and linear algebra as prerequisites. Exposure to elementary ordinary and partial differential equations and calculus of variations are highly desirable, but not absolutely necessary. There are numerous carefully drawn diagrams of geometric figures incorporated throughout the book for illustration and, of course, better understanding. Topological methods are not used in the book, and the concept of manifolds not mentioned, much less treated. So this is an older work that bridges the very foundational and applied aspects of differential geometry with vector analysis, a field and body of knowledge widely used nowadays in the sciences and engineering and exploited in applications such as geodesy. For those insisting on modern approaches and want to omit studying foundations and historical development, please read up on other books such as O'Neill and Spivak.Read more ›
However, Struik can't be used to understand what is happening today. For these purposes,books by O'Neill and do Carmo would be more appropriate. The discussion of manifolds and coordinate charts, the discussion of connection forms, differential forms, covariant derivatives, exterior derivatives, pullbacks and pushforwards can be found in these texts. This is the language of modern geometry.It leads on naturally to tensors, fibre bundles, de Rham cohomology and so on and so forth.The emphasis in modern geometry is on global phenomena, the interaction between local and global (e.g. Morse theory or De Rham cohomology), and the attempt to do everything in an algebraic setting (projective modules, spectral sequences, categories etc.) For this purpose, Struik is useless, though he does have some coverage of forms (he calls them by their earlier name of 'pfaffians').
The price of the book makes it an attractive purchase.
In this regard, this book is no exception. Professor Struik begins at the beginning and goes straight through to the end without skipping any steps and without passing go to collect his $200. He gives the fundamental conceptions of the theory of curves and surfaces, introducing all of the machinery necessary to understand them in a graduated fashion suitable only to the requirements of the topic itself. Elementary calculus will serve the reader well, especially with a smattering of Linear Algebra thrown in. The author wastes no time with sexy side issues or superfluous explanations: Just the basic facts of the fundamental elements here. Those looking for more advanced topics, should consult those books that use this one as their background.
Explanations are sparse, but never deficient; the same is true of the equations. Notation is straightforward and always clear and economical. It is easy to see that (and why) other books on the same topic have used this one as background, but oddly, those other books have been unable to improve upon this one. Other than the fact that the graphics need updating, and more modern topics are missing, this is a splendid effort. Just what I needed.