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Geometry from a Differentiable Viewpoint

2nd Edition

$59.99 (X)

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  • Date Published: October 2012
  • availability: Available
  • format: Paperback
  • isbn: 9780521133111

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  • 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 Hugyens'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, Euclid's geometry of space, further properties of cycloids and map projections, and the use of transformations such as the reflections of the Beltrami disk.

    • Takes historical approach discussing the discovery and construction of non-Euclidean geometry and significant events like Huygens's clock, the mathematics of cartography and Clairaut's relation for geodesics
    • Offers various intertwining approaches to geometry: students begin with the high school synthetic approach and, with development of the differential approach, learn how elementary ideas are related in the new setting
    • Chapter 4 gives a thorough treatment of non-Euclidean geometry, as developed by Lobachevsky and Bolyai, while Chapter 14 parallels this treatment in the differential geometric manner
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    Reviews & endorsements

    Review of the first edition: '… an unusual and interesting account of two subjects and their close historical interrelation.' The Mathematical Gazette

    '… the author has succeeded in making differential geometry an approachable subject for advanced undergraduates.' Andrej Bucki, Mathematical Reviews

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    Product details

    • Edition: 2nd Edition
    • Date Published: October 2012
    • format: Paperback
    • isbn: 9780521133111
    • length: 368 pages
    • dimensions: 253 x 175 x 18 mm
    • weight: 0.63kg
    • contains: 164 b/w illus. 203 exercises
    • availability: Available
  • Table of Contents

    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?

  • Author

    John McCleary, Vassar College, New York
    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.

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