| |
| |
| Preface | |
| |
| |
| Curves in the Plane and in Space | |
| |
| |
| What is a Curve? | |
| |
| |
| Arc-Length | |
| |
| |
| Reparametrization | |
| |
| |
| Level Curves vs. Parametrized Curves | |
| |
| |
| How Much Does a Curve Curve? | |
| |
| |
| Curvature | |
| |
| |
| Plane Curves | |
| |
| |
| Space Curves | |
| |
| |
| Global Properties of Curves | |
| |
| |
| Simple Closed Curves | |
| |
| |
| The Isopcrimetric Inequality | |
| |
| |
| The Four Vertex Theorem | |
| |
| |
| Surfaces in Three Dimensions | |
| |
| |
| What is a Surface? | |
| |
| |
| Smooth Surfaces | |
| |
| |
| Tangents, Normals and Orientability | |
| |
| |
| Examples of Surfaces | |
| |
| |
| Quadric Surfaces | |
| |
| |
| Triply Orthogonal Systems | |
| |
| |
| Applications of the Inverse Function Theorem | |
| |
| |
| The First Fundamental Form | |
| |
| |
| Lengths of Curves on Surfaces | |
| |
| |
| Lsometries of Surfaces | |
| |
| |
| Conformal Mappings of Surfaces | |
| |
| |
| Surface Area | |
| |
| |
| Equiareal Maps and a Theorem of Archimedes | |
| |
| |
| Curvature of Surfaces | |
| |
| |
| The Second Fundamental Form | |
| |
| |
| The Curvature of Curves on a Surface | |
| |
| |
| The Normal and Principal Curvatures | |
| |
| |
| Geometric Interpretation of Pincipal Curvatures | |
| |
| |
| Gaussian Curvature and the Gauss Map | |
| |
| |
| The Gaussian and Mean Curvatures | |
| |
| |
| The Pseudosphere | |
| |
| |
| Flat Surfaces | |
| |
| |
| Surfaces of Constant Mean Curvature | |
| |
| |
| Gaussian Curvature of Compact Surfaces | |
| |
| |
| The Gauss map | |
| |
| |
| Geodesies | |
| |
| |
| Deinition and Basic Properties | |
| |
| |
| Geodesic Equations | |
| |
| |
| Geodesies on Surfaces of Revolution | |
| |
| |
| Geodesies as Shortest Paths | |
| |
| |
| Geodesic Coordinates | |
| |
| |
| Minimal Surfaces | |
| |
| |
| Plateau's Problem | |
| |
| |
| Examples of Minimal Surfaces | |
| |
| |
| Gauss map of a Minimal Surface | |
| |
| |
| Minimal Surfaces and Holomorphic Functions | |
| |
| |
| Gauss's Theorema Egregiuxn | |
| |
| |
| Gauss's Remarkable Theorem | |
| |
| |
| Isomctries of Surfaces | |
| |
| |
| The Codazzi-Mainardi Equations | |
| |
| |
| Compact Surfaces of Constant Gaussian Curvature | |
| |
| |
| The Gauss-Bonnet Theorem | |
| |
| |
| Gauss-Bonnet for Simple Closed Curves | |
| |
| |
| Gauss-Bonnet for Curvilinear Polygons | |
| |
| |
| Gauss-Bonnet for Compact Surfaces | |
| |
| |
| Singularities of Vector Fields | |
| |
| |
| Critical Points | |
| |
| |
| Solutions | |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| Index | |