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Elementary Differential Geometry

Elementary Differential Geometry - 2nd edition

ISBN13: 978-0125267458

Cover of Elementary Differential Geometry 2ND 97 (ISBN 978-0125267458)
ISBN13: 978-0125267458
ISBN10: 0125267452
Cover type:
Edition/Copyright: 2ND 97
Publisher: Academic Press, Inc.
Published: 1997
International: No

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Elementary Differential Geometry - 2ND 97 edition

ISBN13: 978-0125267458

Barrett O'Neill

ISBN13: 978-0125267458
ISBN10: 0125267452
Cover type:
Edition/Copyright: 2ND 97
Publisher: Academic Press, Inc.

Published: 1997
International: No
Summary

Written primarily for readers who have completed the standard first courses in calculus and linear algebra, Elementary Differential Geometry, Second Edition provides an introduction to the geometry of curves and surfaces. Although the popular First Edition has been extensively modified, this Second Edition maintains the elementary character of that volume, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis has been placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard. For readers with access to the symbolic computation programs, Mathematica or Maple, the book includes approximately 30 optional computer exercises. These are not intended as an essential part of the book, but rather an extension. No computer skill is necessary to take full advantage of this comprehensive text.

Gives detailed examples for all essential ideas

Provides more than 300 exercises

Features more than 200 illustrations

Includes an introduction to using computers, and supplies answers to computer exercises given for both Mathematica and Maple systems

Author Bio

O'Neill, Barrett : University of California-Los Angeles

Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.

Table of Contents

Preface.
Introduction.


1. Calculus on Euclidean Space

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

2. Frame Fields

Dot Product.
Curves.
The Frenet Formulas.
Arbitrary Speed Curves.
Covariant Derivatives.
Frame Fields.
Connection Forms.
The Structural Equations.

3. Euclidean Geometry

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

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.

5. Shape Operators

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

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.

7. Riemannian Geometry

Geometric Surfaces.
Gaussian Curvature.
Covariant Derivative.
Geodesics.
Clairaut Parametrizations.
The Gauss-Bonnet Theorem.
Applications of Gauss-Bonnet.

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.

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