.

0 (hodnocen0 x )

(6) Půjčeno:12x 

BK

London ; New York : Springer, c2001

ix, 332 s. : il. ; 24 cm

ISBN 1-85233-152-6 (brož.)

Springer undergraduate mathematics series

Obsahuje rejstřík

Popsáno podle 10. dotisku 2008

000068757

Preface v // 1. Curves in the Plane and in Space I // 1.1 What is a Curve ? 1 // 1.2 Arc-Length 7 // 1.3 Reparametrization 10 // 1.4 Level Curves vs. Parametrized Curves 16 // 2. How Much Does a Curve Curve ? 23 // 2.1 Curvature 23 // 2.2 Plane Curves 28 // 2.3 Space Curves 36 // 3. Global Properties of Curves 47 // 3.1 Simple Closed Curves 47 // 3.2 The Isoperimetric Inequality 51 // 3.3 The Four Vertex Theorem 55 // 4. Surfaces in Three Dimensions 59 // 4.1 What is a Surface ? 59 // 4.2 Smooth Surfaces 66 // 4.3 Tangents, Normals and Orientability 74 // 4.4 Examples of Surfaces 78 // 4.5 Quadric Surfaces 84 // 4.6 Triply Orthogonal Systems 90 // 4.7 Applications of the Inverse Function Theorem 93 // VII // Vlil // Elementary Differential Geometry // 5. The First Fundamental Form 97 // 5.1 Lengths of Curves on Surfaces 97 // 5.2 Isometries of Surfaces 101 // 5.3 Conformal Mappings of Surfaces 106 // 5.4 Surface Area 112 // 5.5 Equiareal Maps and a Theorem of Archimedes 116 // 6. Curvature of Surfaces 123 // 6.1 The Second Fundamental Form 123 // 6.2 The Curvature of Curves on a Surface 127 // 6.3 The Normal and Principal Curvatures 130 // 6.4 Geometric Interpretation of Principal Curvatures 141 // 7. Gaussian Curvature and the Gauss Map 147 // 7.1 The Gaussian and Mean Curvatures 147 // 7.2 The Pseudosphere 151 //

7.3 Flat Surfaces 155 // 7.4 Surfaces of Constant Mean Curvature 161 // 7.5 Gaussian Curvature of Compact Surfaces 164 // 7.6 The Gauss map 165 // 8. Geodesics 171 // 8.1 Definition and Basic Properties 171 // 8.2 Geodesic Equations 175 // 8.3 Geodesics on Surfaces of Revolution 181 // 8.4 Geodesics as Shortest Paths 190 // 8.5 Geodesic Coordinates 197 // 9. Minimal Surfaces 201 // 9.1 Plateau’s Problem 201 // 9.2 Examples of Minimal Surfaces 207 // 9.3 Gauss map of a Minimal Surface 217 // 9.4 Minimal Surfaces and Holomorphic Functions 219 // Gauss’s Theore // a Egregia 229 // Gauss’s Remarkable Theore: // It // 229 // 10.2 Isometries of Surfaces 238 // 10.3 The Codazzi-Mainardi Equations 240 // 10.4 Compact Surfaces of Constant Gaussian Curvature 244 // 11. // The Gauss-Bonnet Theore: // II // 247 // 11.1 Gauss-Bonnet for Simple Closed Curves 247 // 11.2 Gauss-Bonnet for Curvilinear Polygons 252 // 11.3 Gauss-Bonnet for Compact Surfaces 258 // 11.4 Singularities of Vector Fields 269 // 11.5 Critical Points 275 // Solutions 281 // Chapter 1 272 // Chapter 2 275 // Chapter 3 279 // Chapter 4 280 // Chapter 5 286 // Chapter 6 289 // Chapter 7 294 // Chapter 8 299 // Chapter 9 307 // Chapter 10 311 // Chapter 11 314 // Index 329