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Lectures on Classical Differential Geometry: Second Edition [Paperback]

Dirk J. Struik
4.5 out of 5 stars  See all reviews (2 customer reviews)
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Book Description

April 1 1988 0486656098 978-0486656090 Second Edition

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student.
Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there.
For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.


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4.0 out of 5 stars Good treatment of classical differential geometry Oct. 20 2000
By A. Ali
Format:Paperback
Struik's book provides solid coverage of curve and surface theory from the classical point of view, i.e. the kind of stuff Monge, Serret, Frenet and Gauss did. I agree that the book should be on the shelves of mathematicians. A number of classical topics are simply not in vogue these days, and one can find them discussed at length in Struik, or in the exercises. In this sense the book certainly has a more geometric flavor than a number of contemporary texts.
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.
Was this review helpful to you?
Format:Paperback
I simply cannot believe I am the first reviewer of this book!This book should be on the shelf of every mathematician interested in geometry, every computer graphics specialist, everyone interested in solid modelling. For ten bucks, you get a great summary of a wide range of topics in "classical differential geometry" -- the stuff geometers were interested in one hundred years ago. Today it's gauge and string theory -- but the topics discussed in this book are timeless, and many have seen remarkable renaissances in recent years. It is a wonderful little book ... I am using it to teach a basic differential geometry course next year.
Was this review helpful to you?
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com: 4.6 out of 5 stars  5 reviews
33 of 33 people found the following review helpful
5.0 out of 5 stars Very Readable Work on Classical Differential Geometry Feb. 26 2006
By Raymond Woo - Published on Amazon.com
Format:Paperback
While it is quite true Dirk Struik's work is on classical differential geometry, the older methods and treatment do not necesarily imply obsolescence or mediocrity as some readers or reviewers suggest in their evaluations. Classical Analysis is still an important branch of Mathematical Analysis. So classical approaches and topics should not be dismissed as a waste of time, useless, outdated or even invalid. Remember Andrew Wiles' recent attack on Fermat's Last Theorem and his ultimate proof of its validity, an event that made headline news. That is a quintessential classical problem in mathematics (i.e., in number theory), only recently resolved. So remember: CLASSICAL Differential Geometry is part of the title.

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. These are essential to approaching the subject of differential geometry from a more modern and global perspective with heavy emphasis on rigor in proofs and derivations, mathematically speaking. (Also, there are tons of other newer works, i.e., on "modern differential geometry", I am unfamiliar with. They are probably available for browsing in college bookstores.)

The author begins by leading the reader from analytic geometry in 3-dimensions into theory of surfaces, done the old fashion or classical way, i.e., utilizing vector calculus and not much more. Along the way, he takes the reader through subjects such as Euler's theorem, Dupin's indicatrix and various methods for surfaces. Then he continues with developing important fundamental equations underlying surfaces, e.g., Gauss-Weingarten equations, looks at Gauss and Codazzi equations, and proceeds to geodesics and variational methods. He includes a somewhat detailed treatment of the Gauss-Bonnet theorem as he progresses. He ends up with introducing concepts in conformal mapping, which plays an important role in differential geometry, minimal surfaces and various applications, one of which is geodesic mapping useful in geodesy, surveys and map-making. He does all of it with clarity and focus, including problems or "exercises" as he calls it, in under 240 pages - brevity that is rare in many mathematical books and works these days.

For those with a mind for or bent on applications, e.g., applied physics (geophysics), applied mathematics, astronomy, geodesy and aerospace engineering, this book would be an excellent introduction to differential geometry and the classical theories of surfaces - being that one need not worry about abstract analysis and topological aspects of mathematics. Perhaps the title should be "Topics in Classical Differential Geometry" or "Introduction to the Theory of Surfaces in Classical Differential Geometry". But one must keep in mind that Dirk Struik is an old MIT hand and contemporary of Norbert Wiener, also at MIT, and Richard Courant (and many great German-educated mathematicians) who lived and worked in the early to mid-20th century, a long time ago and before computers became commonplace, an era in which total abstraction in mathematics and physics was not quite widely emphasized, but clear concrete thinking was important. A good friend of mine and co-worker who studied at the University of California, Berkeley, told me he had great respect for the classical geometers such as Struik and Eisenhart, understanding that they built ideas from a scratch and wrote in such a way that readers can discern the physical origins of geometry, in particular of differential geometry, a subject that supposedly started with Gauss during the early or mid-19th century when he performed survey work for his government in Germany. (The term "torsion" introduced and sed by Struik in the first few chapters of the book comes from classical mechanics, and is commonly employed in mechanical structures/structural engineering nowadays.)

I for one am an aerospace engineer. There were one or more occasions where I consulted the book for formulas and expressions of curved surfaces and spheroids in my work of flight navigation (flying over the ellipsoidal Earth, as one example). I am sure that are other areas, e.g., space engineering and relativity, where classical methods of differential geometry embodied in Struik's book can come in handy.

The only problem I have with the book is that the "exercises" do not come with solutions, but I do not think that is a major drawback unless one uses it as a textbook for a course that requires assignments and drill exercises.

Judge for yourself by borrowing this book to read, i.e., if you are interested, can tell whether you like or dislike it on the first pass, and for what reasons one way or another. Find out for yourself.
40 of 42 people found the following review helpful
4.0 out of 5 stars Good treatment of classical differential geometry Oct. 20 2000
By A. Ali - Published on Amazon.com
Format:Paperback
Struik's book provides solid coverage of curve and surface theory from the classical point of view, i.e. the kind of stuff Monge, Serret, Frenet and Gauss did. I agree that the book should be on the shelves of mathematicians. A number of classical topics are simply not in vogue these days, and one can find them discussed at length in Struik, or in the exercises. In this sense the book certainly has a more geometric flavor than a number of contemporary texts.
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.
22 of 22 people found the following review helpful
5.0 out of 5 stars Struik's book - a classic on classical differential geometry July 21 2000
By Ronald K. Perline - Published on Amazon.com
Format:Paperback
I simply cannot believe I am the first reviewer of this book!This book should be on the shelf of every mathematician interested in geometry, every computer graphics specialist, everyone interested in solid modelling. For ten bucks, you get a great summary of a wide range of topics in "classical differential geometry" -- the stuff geometers were interested in one hundred years ago. Today it's gauge and string theory -- but the topics discussed in this book are timeless, and many have seen remarkable renaissances in recent years. It is a wonderful little book ... I am using it to teach a basic differential geometry course next year.
9 of 10 people found the following review helpful
5.0 out of 5 stars The most consistent reliable and readable so far May 12 2009
By Herbert L Calhoun - Published on Amazon.com
Format:Paperback|Verified Purchase
With this book, I hope I have finally broken the code and reached a critical mass in advanced mathematical understanding. These Dover Series books allow "it all to hang out." It is "old school" in the best sense of that phrase: that is, in the sense that they do no "sugar coat" their explanations. They do not "dumb it down" or "fancy it up to" ease the pain. One knows what one is up against when one picks up a book from the "Dover Series." They are always clean and sparse in their explanations.

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.

Five Stars.
1 of 10 people found the following review helpful
4.0 out of 5 stars classical Jan. 30 2006
By William Warren - Published on Amazon.com
Format:Paperback
This is a survey of classical i.e. early 20th century differential geometry and not a more "modern" abstract treatment.
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