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Differential Geometry of Curves and Surfaces
     

Differential Geometry of Curves and Surfaces

by Thomas F. Banchoff
 

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ISBN-10: 1568814569

ISBN-13: 9781568814568

Pub. Date: 03/01/2010

Publisher: Taylor & Francis

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

Overview

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.

Product Details

ISBN-13:
9781568814568
Publisher:
Taylor & Francis
Publication date:
03/01/2010
Pages:
352
Product dimensions:
7.50(w) x 9.30(h) x 0.90(d)

Related Subjects

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

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