Elementary Differential GeometrySpringer Science & Business Media, 2001 - 332페이지 Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques. It is a subject that contains some of the most beautiful and profound results in mathematics, yet many of them are accessible to higher level undergraduates.Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times. Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there. |
목차
Curves in the Plane and in Space | 1 |
12 ArcLength | 7 |
13 Reparametrisation | 10 |
14 Level Curves vs Parametrised Curves | 16 |
How Much Does a Curve Curve? | 23 |
22 Plane Curves | 28 |
23 Space Curves | 36 |
Global Properties of Curves | 47 |
Geodesies | 171 |
82 Geodesic Equations | 175 |
83 Geodesies on Surfaces of Revolution | 181 |
84 Geodesics as Shortest Paths | 190 |
85 Geodesic Coordinates | 197 |
Minimal Surfaces | 201 |
92 Examples of Minimal Surfaces | 207 |
93 Gauss Map of a Minimal Surface | 217 |
32 The Isoperimetric Inequality | 51 |
33 The Four Vertex Theorem | 55 |
Surfaces in Three Dimensions | 59 |
42 Smooth Surfaces | 66 |
43 Tangents Normals and Orientability | 74 |
44 Examples of Surfaces | 78 |
45 Quadric Surfaces | 84 |
46 Triply Orthogonal Systems | 90 |
47 Applications of the Inverse Function Theorem | 93 |
The First Fundamental Form | 97 |
52 Isometries of Surfaces | 101 |
53 Conformal Mappings of Surfaces | 106 |
54 Surface Area | 112 |
55 Equiareal Maps and a Theorem of Archimedes | 116 |
Curvature of Surfaces | 123 |
62 The Curvature of Curves on a Surface | 127 |
63 The Normal and Principal Curvatures | 130 |
64 Geometric Interpretation of Principal Curvatures | 141 |
Gaussian Curvature and the Gauss Map | 147 |
72 The Pseudosphere | 151 |
73 Flat Surfaces | 155 |
74 Surfaces of Constant Mean Curvature | 161 |
75 Gaussian Curvature of Compact Surfaces | 164 |
76 The Gauss Map | 166 |
94 Minimal Surfaces and Holomorphic Functions | 219 |
Gausss Theorema Egregium | 229 |
102 Isometries of Surfaces | 238 |
103 The CodazziMainardi Equations | 240 |
104 Compact Surfaces of Constant Gaussian Curvature | 244 |
The GaussBonnet Theorem | 247 |
112 GaussBonnet for Curvilinear Polygons | 252 |
113 GaussBonnet for Compact Surfaces | 258 |
114 Singularities of Vector Fields | 269 |
115 Critical Points | 275 |
Solutions | 281 |
Chapter 2 | 284 |
Chapter 3 | 288 |
Chapter 4 | 289 |
Chapter 5 | 295 |
Chapter 6 | 298 |
Chapter 7 | 303 |
Chapter 8 | 308 |
Chapter 9 | 316 |
Chapter 10 | 319 |
Chapter 11 | 323 |
Index | 329 |
기타 출판본 - 모두 보기
자주 나오는 단어 및 구문
angle arc-length assume catenoid circle contained Corollary cos² cosh cosh² curve in R³ curvey cylinder defined differentiating dot denotes du² dv² Example Exercise Gauss map gaussian curvature generalised geodesic equations gives helicoid Hence holomorphic integral intersect inverse function theorem isometry K₁ matrix mean curvature minimal surface never zero non-zero open set open subset parallel parameter perpendicular plane curve principal curvatures proof Proposition pseudosphere quadric radius regular rigid motion rotation S₁ scalar second fundamental forms Show signed curvature simple closed curve sin² sinh sint smooth function standard unit normal straight line surface of revolution surface patch t₁ tangent plane tangent vector torsion triangulation unit sphere unit vector unit-speed curve vector field vertex xy-plane z-axis θυ σ₁ ση συν Χ συ ди