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Geometry from a Differentiable Viewpoint, by John McCleary
Download Geometry from a Differentiable Viewpoint, by John McCleary
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This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development. The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane. While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds.
- Sales Rank: #2419263 in Books
- Brand: Brand: Cambridge University Press
- Published on: 1995-01-27
- Original language: English
- Number of items: 1
- Dimensions: 9.72" h x .67" w x 6.85" l, 1.25 pounds
- Binding: Paperback
- 324 pages
- Used Book in Good Condition
Review
"...the author has succeeded in making differential geometry an approachable subject for advanced undergraduates." Andrej Bucki, Mathematical Reviews
About the Author
John McCleary is Professor of Mathematics at Vassar College on the Elizabeth Stillman Williams Chair. His research interests lie at the boundary between geometry and topology, especially where algebraic topology plays a role. His papers on topology have appeared in Inventiones Mathematicae, the American Journal of Mathematics and other journals, and he has written expository papers that have appeared in American Mathematical Monthly. He is also interested in the history of mathematics, especially the history of geometry in the nineteenth century and of topology in the twentieth century. He is the author of A User's Guide to Spectral Sequences and A First Course in Topology: Continuity and Dimension and he has edited proceedings in topology and in history, as well as a volume of the collected works of John Milnor. He has been a visitor to the mathematics institutes in Goettingen, Strasbourg and Cambridge, and to MSRI in Berkeley.
Most helpful customer reviews
2 of 2 people found the following review helpful.
A Bridge and Foundation Stone
By G. A. Schoenagel
Previous reviews of this publication prompt my own exegesis.
The author intimates that an acquaintance with Multivariate Calculus plus exposure to elementary Linear Algebra and Analysis
should be adequate preparation to follow much of the material. This, in my view, is absolutely correct.
With those prerequisites,the stated purpose of preparing the reader for further coursework in modern differential geometry is achieved.
Three Major Parts comprise the book:
(1) Introductory and Spherical Geometry
(2) Curves,Surfaces,Curvature
(3) Recapitulation of the first two parts with introduction to abstract surfaces.
The first part presents essentials of Spherical Geometry, Euclid's Postulates and onward to Non-Euclidean Geometry.
Part One is very accessible and should present the diligent reader with no difficulties.
The second part provides study of Curves and Surfaces.The material is more demanding, as is appropriate.
The Final part provide a panoramic view of, and introduction to, manifolds.
Progression from part one to two is easy, from part two to three is much more demanding. Part three then falls into place nicely!
Before attempting this book, consider that firm background in high school geometry and (at least) a few terms of Calculus are necessary.
(Assimilating the Geometry and Calculus text's of Moise, on both accounts, should suffice as preparation.)
Assuming the aforementioned background, this text offers a marvelous choice for further expatiation.
The exercises are straightforward. Solutions to some of the exercises are included.
Altogether, a satisfying, if not ambitious, attempt to whet the appetite for future study of differential geometry.
The exposition is motivated by history as preparation for modern developments.
A rather enjoyable journey.
20 of 21 people found the following review helpful.
not for the uninitiated
By Shawn McDougal
I'm a master's student in math. I bought the book thinking I'd use it for an independent study. I was wrong.
The book has interesting historical tidbits and some classical proofs, including material I hadn't seen elsewhere. However, it takes little time to explain to the novice exactly what's going on. It comes off more as a set of lecture notes than as a text for self-study.
For instance, in ch. 8 McCleary breezes through the basics of regular surfaces--coordinate charts, differentiability, implicit/inverse function theorem, the tangent space, orientability, the first fundamental form in about 19 pages. This is the same foundational material that folk like do Carmo or O'Neill rightfully spend 60-70 pages to cover.
His treatment of the Gauss map and the second fundamental form is even more schematic.
If I hadn't already worked the other books, when I got to McCleary's treatment of surfaces I would've been completely lost.
This book is best for people who know basic differential geometry already but are curious about certain historical aspects of it, not for people who are trying to learn differential geometry.
6 of 7 people found the following review helpful.
great history of geometry book, terrible introductory differential geometry book
By Malcolm
Do not buy this inappropriately titled book if you are seeking an introductory text to learn differential geometry. It's not that the concepts in the book are so advanced, so much as not that much space is actually devoted to the subject. The author's real objective is to trace the development of geometry from Euclid to the (relatively) modern formulation of differential geometry, and as a book on that topic it succeeds admirably.
The core theme of the book is that efforts to prove the parallel postulate, or, equivalently, show that non-Euclidean geometries are impossible, inadvertently, through their failure, led to the discovery of many fascinating areas of mathematics, such as hyperbolic and Riemannian geometries, and to the development of philosophical ideas about what actually constitutes mathematics and how it is independent from physical reality. The book culminates with the results of Beltrami and Poincare that showed that hyperbolic and Euclidean geometries are logically equivalent, in the sense that if there is a self-contradiction in one then the other is also impossible, thus putting an end to all attempts to disprove hyperbolic geometry. (Unfortunately, Marilyn vos Savant is unaware of this, or at least she was when she wrote an article some years back criticising Andrew Wiles's proof of Fermat's last theorem because it used hyperbolic geometry.)
As an appendix, McCleary adds a translation of Riemann's lecture "On the hypothesis which lie at the foundations of geometry," perhaps the most influential single lecture in the history of mathematics (and physics), in which, in the mid-1860s, he presented to a general faculty a talk (involving only a single equation) on the foundations of geometry that anticipated the concepts of a manifold and Riemannian geometry as well as general relativity and even hinted at quantum mechanics.
I used this text as a primary reference when conducting an undergraduate seminar on the history of hyperbolic geometry 12 years ago. For this purpose it was suited perfectly, but if you want to learn differential geometry by all means buy one of do Carmo's books or Gallot, Hulin, and LaFontaine.
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