Preface v
Curves in the Plane and in Space 1 (22)
What is a Curve? 1 (6)
Arc-Length 7 (3)
Reparametrization 10 (6)
Level Curves vs. Parametrized Curves 16 (7)
How Much Does a Curve Curve? 23 (24)
Curvature 23 (5)
Plane Curves 28 (8)
Space Curves 36 (11)
Global Properties of Curves 47 (12)
Simple Closed Curves 47 (4)
The Isoperimetric Inequality 51 (4)
The Four Vertex Theorem 55 (4)
Surfaces in Three Dimensions 59 (38)
What is a Surface? 59 (7)
Smooth Surfaces 66 (8)
Tangents, Normals and Orientability 74 (4)
Examples of Surfaces 78 (6)
Quadric Surfaces 84 (6)
Triply Orthogonal Systems 90 (3)
Applications of the Inverse Function Theorem 93 (4)
The First Fundamental Form 97 (26)
Lenghts of Curves on Surfaces 97 (4)
Isometries of Surfaces 101(5)
Conformal Mappings of Surfaces 106(6)
Surface Area 112(4)
Equiareal Maps and a Theorem of Archimedes 116(7)
Curvature of Surfaces 123(24)
The Second Fundamental Form 123(4)
The Curvature of Curves on a Surface 127(3)
The Normal and Principle Curvatures 130(11)
Geometric Interpretation of Principle 141(6)
Curvatures
Gaussian Curvature and the Gauss Map 147(24)
The Gaussian and Mean Curvatures 147(4)
The Pseudosphere 151(4)
Flat Surfaces 155(6)
Surfaces of Constant Mean Curvature 161(3)
Gaussian Curvature of Compact Surfaces 164(1)
The Gauss map 165(6)
Geodesics 171(30)
Definition and Basic Properties 171(4)
Geodesic Equations 175(6)
Geodesics on Surfaces of Revolution 181(9)
Geodesics as Shortest Paths 190(7)
Geodesic Coordinates 197(4)
Minimal Surfaces 201(28)
Plateau's Problem 201(6)
Examples of Minimal Surfaces 207(10)
Gauss map of a Minimal Surface 217(2)
Minimal Surfaces and Holomorphic Functions 219(10)
Gauss's Theorema Egregium 229(18)
Gauss's Remarkable Theorem 229(9)
Isometries of Surfaces 238(2)
The Codazzi-Maiardi Equations 240(4)
Compact Surfaces of Constant Gaussian 244(3)
Curvature
The Gauss-Bonnet Theorem 247(34)
Gauss-Bonnet for Simple Closed Curves 247(5)
Gauss-Bonnet for Curvilinear Polygons 252(6)
Gauss-Bonnet for Compact Surfaces 258(11)
Singularities of Vector Fields 269(6)
Critical Points 275(6)
Solutions 281(48)
Chapter 1 272(3)
Chapter 2 275(4)
Chapter 3 279(1)
Chapter 4 280(6)
Chapter 5 286(3)
Chapter 6 289(5)
Chapter 7 294(5)
Chapter 8 299(8)
Chapter 9 307(4)
Chapter 10 311(3)
Chapter 11 314(15)
Index 329
Curves in the Plane and in Space 1 (22)
What is a Curve? 1 (6)
Arc-Length 7 (3)
Reparametrization 10 (6)
Level Curves vs. Parametrized Curves 16 (7)
How Much Does a Curve Curve? 23 (24)
Curvature 23 (5)
Plane Curves 28 (8)
Space Curves 36 (11)
Global Properties of Curves 47 (12)
Simple Closed Curves 47 (4)
The Isoperimetric Inequality 51 (4)
The Four Vertex Theorem 55 (4)
Surfaces in Three Dimensions 59 (38)
What is a Surface? 59 (7)
Smooth Surfaces 66 (8)
Tangents, Normals and Orientability 74 (4)
Examples of Surfaces 78 (6)
Quadric Surfaces 84 (6)
Triply Orthogonal Systems 90 (3)
Applications of the Inverse Function Theorem 93 (4)
The First Fundamental Form 97 (26)
Lenghts of Curves on Surfaces 97 (4)
Isometries of Surfaces 101(5)
Conformal Mappings of Surfaces 106(6)
Surface Area 112(4)
Equiareal Maps and a Theorem of Archimedes 116(7)
Curvature of Surfaces 123(24)
The Second Fundamental Form 123(4)
The Curvature of Curves on a Surface 127(3)
The Normal and Principle Curvatures 130(11)
Geometric Interpretation of Principle 141(6)
Curvatures
Gaussian Curvature and the Gauss Map 147(24)
The Gaussian and Mean Curvatures 147(4)
The Pseudosphere 151(4)
Flat Surfaces 155(6)
Surfaces of Constant Mean Curvature 161(3)
Gaussian Curvature of Compact Surfaces 164(1)
The Gauss map 165(6)
Geodesics 171(30)
Definition and Basic Properties 171(4)
Geodesic Equations 175(6)
Geodesics on Surfaces of Revolution 181(9)
Geodesics as Shortest Paths 190(7)
Geodesic Coordinates 197(4)
Minimal Surfaces 201(28)
Plateau's Problem 201(6)
Examples of Minimal Surfaces 207(10)
Gauss map of a Minimal Surface 217(2)
Minimal Surfaces and Holomorphic Functions 219(10)
Gauss's Theorema Egregium 229(18)
Gauss's Remarkable Theorem 229(9)
Isometries of Surfaces 238(2)
The Codazzi-Maiardi Equations 240(4)
Compact Surfaces of Constant Gaussian 244(3)
Curvature
The Gauss-Bonnet Theorem 247(34)
Gauss-Bonnet for Simple Closed Curves 247(5)
Gauss-Bonnet for Curvilinear Polygons 252(6)
Gauss-Bonnet for Compact Surfaces 258(11)
Singularities of Vector Fields 269(6)
Critical Points 275(6)
Solutions 281(48)
Chapter 1 272(3)
Chapter 2 275(4)
Chapter 3 279(1)
Chapter 4 280(6)
Chapter 5 286(3)
Chapter 6 289(5)
Chapter 7 294(5)
Chapter 8 299(8)
Chapter 9 307(4)
Chapter 10 311(3)
Chapter 11 314(15)
Index 329