Elementary Differential Geometry, Revised 2nd Edition

EL an an an an Wr an Wr EL Wr EL Wr EL Wr. ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.

The Second Edition maintains the accessibility of the first, while providing an introduction to the use of computers. The emphasis was placed on the topological properties, properties of geodesics, singularities of vector fields, and the theories of Bonnet and Hadamard.

This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs. Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this content is a comprehensive text.

* Fortieth anniversary of publication! Over 36,000 copies sold worldwide
* Accessible, practical yet rigorous approach to a complex topic - also suitable for self-study
* Extensive update of appendices on Mathematica and Maple software packages
* Thorough streamlining of the second edition's numbering system
* Fuller information on solutions to odd-numbered problems
* Additional exercises and hints guide students in using the latest computer modeling tools

Preface

Introduction

Chapter 1: Calculus on Euclidean Space:
Euclidean Space. Tangent Vectors. Directional Derivatives. Curves in R3. 1-forms. Differential Forms. Mappings.

Chapter 2: Frame Fields:
Dot Product. Curves. The Frenet Formulas. ArbitrarySpeed Curves. Covariant Derivatives. Frame Fields. Connection Forms. The Structural Equations.

Chapter 3: Euclidean Geometry:
Isometries of R3. The Tangent Map of an Isometry. Orientation. Euclidean Geometry. Congruence of Curves.

Chapter 4: Calculus on a Surface:
Surfaces in R3. Patch Computations. Differentiable Functions and Tangent Vectors. Differential Forms on a Surface. Mappings of Surfaces. Integration of Forms. Topological Properties. Manifolds.

Chapter 5: Shape Operators:
The Shape Operator of M R3. Normal Curvature. Gaussian Curvature. Computational Techniques. The Implicit Case. Special Curves in a Surface. Surfaces of Revolution.

Chapter 6: Geometry of Surfaces in R3:
The Fundamental Equations. Form Computations. Some Global Theorems. Isometries and Local Isometries. Intrinsic Geometry of Surfaces in R3. Orthogonal Coordinates. Integration and Orientation. Total Curvature. Congruence of Surfaces.

Chapter 7: Riemannian Geometry: Geometric Surfaces. Gaussian Curvature. Covariant Derivative. Geodesics. Clairaut Parametrizations. The Gauss-Bonnet Theorem. Applications of Gauss-Bonnet.

Chapter 8: Global Structures of Surfaces: Length-Minimizing Properties of Geodesics. Complete Surfaces. Curvature and Conjugate Points. Covering Surfaces. Mappings that Preserve Inner Products. Surfaces of Constant Curvature. Theorems of Bonnet and Hadamard.

Appendix

Bibliography

Answers to Odd-Numbered Exercises

Subject Index