Differential Geometry of Curves and Surfaces

Differential Geometry of Curves and Surfaces

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Summary

Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). Some of the more interesting theorems explore relationships between local and global properties.

A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena.

Table of Contents

Preface

Acknowledgements

  1. Plane Curves: Local Properties
  2. Parameterizations

    Position, Velocity, and Acceleration

    Curvature

    Osculating Circles, Evolutes, and Involutes

    Natural Equations

  3. Plane Curves: Global Properties
  4. Basic Properties

    Rotation Index

    Isoperimetric Inequality

    Curvature, Convexity, and the Four-Vertex Theorem

  5. Curves in Space: Local Properties
  6. Definitions, Examples, and Differentiation

    Curvature, Torsion, and the Frenet Frame

    Osculating Plane and Osculating Sphere

    Natural Equations

  7. Curves in Space: Global Properties
  8. Basic Properties

    Indicatrices and Total Curvature

    Knots and Links

  9. Regular Surfaces
  10. Parametrized Surfaces

    Tangent Planes and Regular Surfaces

    Change of Coordinates

    The Tangent Space and the Normal Vector

    Orientable Surfaces

  11. The First and Second Fundamental Forms
  12. The First Fundamental Form

    The Gauss Map

    The Second Fundamental Form

    Normal and Principal Curvatures

    Gaussian and Mean Curvature

    Ruled Surfaces and Minimal Surfaces

  13. The Fundamental Equations of Surfaces
  14. Tensor Notation

    Gauss’s Equations and the Christoffel Symbols

    Codazzi Equations and the Theorema Egregium

    The Fundamental Theorem of Surface Theory

  15. Curves on Surfaces

Curvatures and Torsion

Geodesics

Geodesic Coordinates

Gauss-Bonnet Theorem and Applications

Intrinsic Geometry

Bibliography

Author Bio(s)

Thomas F. Banchoff is a geometer and has been a professor at Brown University since 1967. Banchoff was president of the MAA from 1999-2000. He is published widely and known to a broad audience as editor and commentator on Abbotts Flatland. He has been the recipient of such awards as the MAA National Award for Distinguished College or University Teaching of Mathematics and most recently the 2007 Teaching with Technology Award.

Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.

Editorial Reviews

… a complete guide for the study of classical theory of curves and surfaces and is intended as a textbook for a one-semester course for undergraduates …The main advantages of the book are the careful introduction of the concepts, the good choice of the exercises, and the interactive computer graphics, which make the text well-suited for self-study. …The access to online computer graphics applets that illustrate many concepts and theorems presented in the text provides the readers with an interesting and visually stimulating study of classical differential geometry. … I strongly recommend [this book and Differential Geometry of Manifolds] to anyone wishing to enter into the beautiful world of the differential geometry.
—Velichka Milousheva, Journal of Geometry and Symmetry in Physics, 2012

Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book … Some of the more interesting theorems explore relationships between local and global properties. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena.
L’Enseignement Mathématique (2) 57 (2011)

… an intuitive and visual introduction to the subject is beneficial in an undergraduate course. This attitude is reflected in the text. The authors spent quite some time on motivating particular concepts and discuss simple but instructive examples. At the same time, they do not neglect rigour and precision. … As a distinguishing feature to other textbooks, there is an accompanying web page containing numerous interactive Java applets. … The applets are well-suited for use in classroom teaching or as an aid to self-study.
—Hans-Peter Schröcker, Zentralblatt MATH 1200

Coming from intuitive considerations to precise definitions the authors have written a very readable book. Every section contains many examples, problems and figures visualizing geometric properties. The understanding of geometric phenomena is supported by a number of available Java applets. This special feature distinguishes the textbook from others and makes it recommendable for self studies too. … highly recommendable …
—F. Manhart, International Mathematical News, August 2011

… the authors succeeded in making this modern view of differential geometry of curves and surfaces an approachable subject for advanced undergraduates.
—Andrew Bucki, Mathematical Reviews, Issue 2011h

… an essential addition to academic library Mathematical Studies instructional reference collections, as well as an ideal classroom textbook.
Midwest Book Review, May 2011

Downloads Updates

Resource OS Platform Updated Description Instructions
Cross Platform January 17, 2013 Web Link for Applets click on http://diffgeo.akpeters.com/
K00382.PDF Cross Platform August 22, 2011 Errata

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