O'Neill Elementary Differential Geometry

1. Auflage 2014
ISBN: 978-1-4832-6811-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 422 Seiten, Web PDF

Elementary Differential Geometry focuses on the elementary account of the geometry of curves and surfaces. The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry. The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation. The text is a valuable reference for students interested in elementary differential geometry.

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.

Autoren/Hrsg.

O'Neill, Barrett

Inhaltsverzeichnis

1;Front Cover;1
2;Elementary Differential Geometry;4
3;Copyright Page;5
4;Table of Contents;8
5;Preface;6
6;Introduction;12
7;Chapter I. Calculus on Euclidean Space;14
7.1;1 Euclidean Space;14
7.2;2 Tangent Vectors;17
7.3;3 Directional Derivatives;22
7.4;4 Curves in E;26
7.5;5 1-Forms;33
7.6;6 Differential Forms;37
7.7;7 Mappings;43
7.8;8 Summary;52
8;Chapter II. Frame Fields;53
8.1;1 Dot Product;53
8.2;2 Curves;62
8.3;3 The Frenet Formulas;67
8.4;4 Arbitrary-Speed Curves;77
8.5;5 Covariant Derivatives;88
8.6;6 Frame Fields;92
8.7;7 Connection Forms;96
8.8;8 The Structural Equations;102
8.9;9 Summary;107
9;Chapter III. Euclidean Geometry;109
9.1;1 Isometries of E;109
9.2;2 The Derivative Map of an Isometry;115
9.3;3 Orientation;118
9.4;4 Euclidean Geometry;123
9.5;5 Congruence of Curves;127
9.6;6 Summary;134
10;Chapter IV. Calculus on a Surface;135
10.1;1 Surfaces in E;135
10.2;2 Patch Computations;144
10.3;3 Differentiable Functions and Tangent Vectors;154
10.4;4 Differential Forms on a Surface;163
10.5;5 Mappings of Surfaces;169
10.6;6 Integration of Forms;178
10.7;7 Topological Properties of Surfaces;187
10.8;8 Manifolds;193
10.9;9 Summary;198
11;Chapter V. Shape Operators;200
11.1;1 The Shape Operator of M c E;200
11.2;2 Normal Curvature;206
11.3;3 Gaussian Curvature;214
11.4;4 Computational Techniques;221
11.5;5 Special Curves in a Surface;234
11.6;6 Surfaces of Revolution;245
11.7;7 Summary;255
12;Chapter VI. Geometry of Surfaces in E;256
12.1;1 The Fundamental Equations;256
12.2;2 Form Computations;262
12.3;3 Some Global Theorems;267
12.4;4 Isometries and Local Isometries;274
12.5;5 Intrinsic Geometry of Surfaces in E;282
12.6;6 Orthogonal Coordinates;287
12.7;7 Integration and Orientation;291
12.8;8 Congruence of Surfaces;308
12.9;9 Summary;314
13;Chapter VII. Riemannian Geometry;315
13.1;1 Geometric Surfaces;315
13.2;2 Gaussian Curvature;321
13.3;3 Covariant Derivative;329
13.4;4 Geodesies;337
13.5;5 Length-Minimizing Properties of Geodesies;350
13.6;6 Curvature and Conjugate Points;363
13.7;7 Mappings that Preserve Inner Products;373
13.8;8 The Gauss-Bonnet Theorem;383
13.9;9 Summary;400
14;Bibliography;402
15;Answers to Odd-Numbered Exercises;404
16;INDEX;416