Geometry from a differentiable viewpoint / John McCleary.

Format:

Book

Author/Creator:

McCleary, John, 1952- author.

Language:

English

Subjects (All):

Geometry, Differential.

Physical Description:

1 online resource (xv, 357 pages) : digital, PDF file(s).

Edition:

Second edition.

Place of Publication:

Cambridge : Cambridge University Press, 2013.

Language Note:

English

Summary:

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. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. 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. The presentation is enlivened by historical diversions such as Huygens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics and the use of transformations such as the reflections of the Beltrami disk.

Contents:

Machine generated contents note: Part I. Prelude and Themes: Synthetic Methods and Results: 1. Spherical geometry; 2. Euclid; 3. The theory of parallels; 4. Non-Euclidean geometry; Part II. Development: Differential Geometry: 5. Curves in the plane; 6. Curves in space; 7. Surfaces; 8. Curvature for surfaces; 9. Metric equivalence of surfaces; 10. Geodesics; 11. The Gauss-Bonnet Theorem; 12. Constant-curvature surfaces; Part III. Recapitulation and Coda: 13. Abstract surfaces; 14. Modeling the non-Euclidean plane; 15. Epilogue: where from here?.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and indexes.

ISBN:

1-139-79301-2

1-316-08860-X

1-107-25318-7

1-139-02224-5

1-139-77863-3

1-139-77559-6

1-139-78162-6

1-283-71559-7

1-139-77711-4

OCLC:

815823809