Geometry from a differentiable viewpoint mccleary john
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The subject is treated deductively based on definitions, theorems their authors are mostly indicated , proofs often near the original ones. 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. Part B consists of nine chapters and begins with Chapter 6, which presents the theory of plane curves in such a way as to give the basic general motivations once for all for the underlying concepts of differential geometry so that the concepts can be introduced without much motivation in the more complicated cases of space curves and surfaces. This often poses a problem for undergraduates: which direction should be followed? Closely following is a chapter on map and map projections â€” an immediate application of the ideas and techniques from the theory of surfaces. Metric equivalence of surfaces; 10.

The book begins with the theorems of non-Euclidean geometry, then introduces the methods of differential geometry and develops them towards the goal of constructing models of the hyperbolic plane. The theory of parallels; 4. The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss and Riemann is a story that is often broken into parts - axiomatic geometry, non-Euclidean geometry and differential geometry. Chapters 9 and 10 develop the analogue of curvature for curves on a surface, an intrinsic feature of the surface, the Gaussian curvature. The resolution comes in Part C.

The dust jacket for hard covers may not be included. This is the second edition of a fresh look at introductory differential geometry for undergraduates. Reviews: From the Mathematical Reviews: Differential geometry is a subject of basic importance for all mathematicians, regardless of their special interests, and it also furnishes essential ideas and tools to physicists and engineers. The theory of parallels; 4. This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates.

About this Item: Cambridge University Press, 1995. The theory of parallels 4. Pages and cover are clean and intact. This book is an attempt to carry the reader from the familiar Euclidean world to the recent state of development of differential geometry. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. In recent years, it has become a luxury to offer a course in differential geometry in an undergraduate curriculum. This book is designed to make differential geometry an approachable subject for advanced undergraduates.

This book is an appealing introduction to differential geometry that is fully accessible to undergraduates. What is the big picture to which these parts belong? New topics include Clairaut's relation for geodesics and the use of transformations such as the reflections of the Beltrami disk. The book begins with the theorems of non-Euclidean geometry, then introduces the methods of differential geometry and develops them towards the goal of constructing models of the hyperbolic plane. Chapters 4 and 5 deal with an exposition of synthetic non-Euclidean geometry as introduced by Saccheri, Bolyai, and Lobachevskii. The historical stories contain several well-known details, e. The theory of parallels 4. The theory of parallels; 4.

Part C departs some from the usual course of introductory material. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. 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. A number of problems are formulated at the end of the chapters. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. 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.

In a very lively manner the spherical and hyperbolic geometries, the classical theory of curves and surfaces and a great part of Riemannian geometry are presented, as well as some applications the tautochrone and accurate clock of Huygens, map projections and mathematical cartography, Lorentz manifolds as space-time models. 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. Prelude and Themes: Synthetic Methods and Results: 1. Peterson Anticipation of Mainardi- Codazzi equations. 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 author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane. Book is in Used-Good condition.

Metric equivalence of surfaces; 11. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of the non-Euclidean plane. The book consists of three parts it is written in sonata-allegro form. This often poses a problem for undergraduates: which direction should be followed? Synthetic Methods and Results 1. Metric equivalence of surfaces 11. His research interests lie at the boundary between geometry and topology, especially where algebraic topology plays a role.

About this Item: Cambridge University Press, 1995. This is mostly a standard treatment, but it is clear and well-written. His goal is to integrate the pieces, to make the connections, and to take the student from the familiar neighborhood of Euclidean geometry to the state of development of differential geometry at the beginning of the twentieth century. The main theorems of non-Euclidean geometry are presented along with their historical development. The main theorems of non-Euclidean geometry are presented along with their historical development. All this is alternated with historical stories and linking comments. A lot of additional information is contained in the exercises which are given after each of the 17 chapters in total 167.

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. May not contain Access Codes or Supplements. This thoroughly revised second edition includes numerous new exercises and a new solution key. Metric equivalence of surfaces 10. What do these ideas have to do with geometry? The presentation is enlivened by historical diversions such as Huygens's clock and the mathematics of cartography. Interesting diversions are offered, such as Huygens' pendulum clock and mathematical cartography; however, the focus of the book is on the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. 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.