Elementary Differential Geometry, Revised 2/e
暫譯: 初等微分幾何（修訂版第二版）
Barrett O'Neill
- 出版商: Academic Press
- 出版日期: 2006-02-01
- 售價: $3,650
- 貴賓價: 9.5 折 $3,468
- 語言: 英文
- 頁數: 520
- 裝訂: Hardcover
- ISBN: 0120887355
- ISBN-13: 9780120887354
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Description
Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.
The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.
This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.
Preface
Introduction
Chapter 1: Calculus on Euclidean Space:
Euclidean Space. Tangent Vectors. Directional Derivatives. Curves in R3. 1-forms. Differential Forms. Mappings.
Chapter 2: Frame Fields:
Dot Product. Curves. The Frenet Formulas. ArbitrarySpeed Curves. Covariant Derivatives. Frame Fields. Connection Forms. The Structural Equations.
Chapter 3: Euclidean Geometry:
Isometries of R3. The Tangent Map of an Isometry. Orientation. Euclidean Geometry. Congruence of Curves.
Chapter 4: Calculus on a Surface:
Surfaces in R3. Patch Computations. Differentiable Functions and Tangent Vectors. Differential Forms on a Surface. Mappings of Surfaces. Integration of Forms. Topological Properties. Manifolds.
Chapter 5: Shape Operators:
The Shape Operator of M R3. Normal Curvature. Gaussian Curvature. Computational Techniques. The Implicit Case. Special Curves in a Surface. Surfaces of Revolution.
Chapter 6: Geometry of Surfaces in R3:
The Fundamental Equations. Form Computations. Some Global Theorems. Isometries and Local Isometries. Intrinsic Geometry of Surfaces in R3. Orthogonal Coordinates. Integration and Orientation. Total Curvature. Congruence of Surfaces.
Chapter 7: Riemannian Geometry: Geometric Surfaces. Gaussian Curvature. Covariant Derivative. Geodesics. Clairaut Parametrizations. The Gauss-Bonnet Theorem. Applications of Gauss-Bonnet.
Chapter 8: Global Structures of Surfaces: Length-Minimizing Properties of Geodesics. Complete Surfaces. Curvature and Conjugate Points. Covering Surfaces. Mappings that Preserve Inner Products. Surfaces of Constant Curvature. Theorems of Bonnet and Hadamard.
Appendix
Bibliography
Answers to Odd-Numbered Exercises
Subject Index
商品描述(中文翻譯)
**描述**
本書主要為已完成微積分和線性代數標準初級課程的學生所撰寫，《初等微分幾何（修訂第二版）》提供了曲線和曲面的幾何學介紹。
第二版保持了第一版的可讀性，同時引入了計算機的使用並擴展了某些主題的討論。進一步強調了拓撲性質、測地線的性質、向量場的奇異性，以及Bonnet和Hadamard的定理。
本修訂版的第二版對於符號計算程序Mathematica或Maple的命令進行了徹底更新，並增加了額外的計算機練習。與第二版一樣，這些材料補充了內容，但不需要計算機技能即可充分利用這本全面的文本。
**目錄**
前言
介紹
第一章：歐幾里得空間上的微積分：
歐幾里得空間。切向量。方向導數。R3中的曲線。1-形式。微分形式。映射。
第二章：框架場：
點積。曲線。Frenet公式。任意速度曲線。協變導數。框架場。連接形式。結構方程。
第三章：歐幾里得幾何：
R3的等距變換。等距變換的切映射。方向。歐幾里得幾何。曲線的全等性。
第四章：曲面上的微積分：
R3中的曲面。補丁計算。可微函數和切向量。曲面上的微分形式。曲面的映射。形式的積分。拓撲性質。流形。
第五章：形狀算子：
M R3的形狀算子。法向曲率。高斯曲率。計算技術。隱式情況。曲面上的特殊曲線。旋轉曲面。
第六章：R3中曲面的幾何：
基本方程。形式計算。一些全局定理。等距變換和局部等距變換。R3中曲面的內在幾何。正交坐標。積分和方向。總曲率。曲面的全等性。
第七章：黎曼幾何：幾何曲面。高斯曲率。協變導數。測地線。Clairaut參數化。高斯-Bonnet定理。高斯-Bonnet的應用。
第八章：曲面的全局結構：測地線的最小長度性質。完整曲面。曲率和共軛點。覆蓋曲面。保持內積的映射。恆定曲率的曲面。Bonnet和Hadamard的定理。
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