Elementary Differential Geometry

Автор(ы):O'neil Barrett
06.10.2007
Год изд.:1966
Описание: This book is an elementary account of the geometry of curves and surfaces. It is written for students who have completed standard first courses in calculus and linear algebra, and its aim is to introduce some of the main ideas of differential geometry. The traditional undergraduate course in differential geometry has changed very little in the last few decades. By contrast, geometry has been advancing very rapidly at the research level, and there is general agreement that the undergraduate course needs to be brought up to date. The author tried to think through the classical material, to prune and augment it, and to write down the results in a reasonably clean and modern mathematical style. However, he has used a new idea only if it really pays its way by simplifying and clarifying the exposition.
Оглавление:
Elementary Differential Geometry — обложка книги. Обложка книги.
Chapter I. Calculus on Euclidean Space
  1. Euclidean Space [3]
  2. Tangent Vectors [6]
  3. Directional Derivatives [11]
  4. Curves in (?) [15]
  5. 1-Forms [22]
  6. Differential Forms [26]
  7. Mappings [32]
  8. Summary [41]
Chapter II. Frame Fields
  1. Dot Product [42]
  2. Curves [51]
  3. The Freuet Formulas [56]
  4. Arbitrary-Speed Curves [66]
  5. Covariant Derivatives [77]
  6. Frame Fields [81]
  7. Connection Forms [85]
  8. The Structural Equations [91]
  9. Summary [96]
Chapter III. Euclidean Geometry
  1. Isometries of (?) [98]
  2. The Derivative Map of an Isometry [104]
  3. Orientation [107]
  4. Euclidean Geometry [112]
  5. Congruence of Curves [116]
  6. Summary [123]
Chapter IV. Calculus on a Surface
  1. Surfaces in (?) [124]
  2. Patch Computations [133]
  3. Differentiable Functions and Tangent Vectors [143]
  4. Differential Forms on a Surface [152]
  5. Mappings of Surfaces [158]
  6. Integration of Forms [167]
  7. Topological Properties of Surfaces [176]
  8. Manifolds [182]
  9. Summary [187]
Chapter V. Shape Operators
  1. The Shape Operator of (?) [189]
  2. Normal Curvature [195]
  3. Gaussian Curvature [203]
  4. Computational Techniques [210]
  5. Special Curves in a Surface [223]
  6. Surfaces of Revolution [234]
  7. Summary [244]
Chapter VI. Geometry of Surfaces in (?)
  1. The Fundamental Equations [245]
  2. Form Computations [251]
  3. Some Global Theorems [256]
  4. Isometries and Local Isometries [263]
  5. Intrinsic Geometry of Surfaces in E3 [271]
  6. Orthogonal Coordinates [270]
  7. Integration and Orientation [280]
  8. Congruence of Surfaces [297]
  9. Summary [303]
Chapter VII. Riemannian Geometry
  1. Geometric Surfaces [304]
  2. Gaussian Curvature [310]
  3. Covariant Derivative [318]
  4. Geodesies [326]
  5. Length-Minimizing Properties of Geodesies [339]
  6. Curvature and Conjugate Points [352]
  7. Mappings that Preserve Inner Products [362]
  8. The Gauss-Bonnet Theorem [372]
  9. Summary [389]
BIBLIOGRAPHY [391]
Answers то Odd-Numbeked Exbkcises [393]
INDEX [405]
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Язык:ENG
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