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內容簡介

　　微分幾何基礎講述的是麯綫和平麵的微分幾何學的主要結論適閤於本科生第一學期的課程。《微分幾何基礎（第2版 英文版）》在改版中有如下新的特徵：有一章專門講述非歐幾何，該課題在數學史上具有重要的影響且對現代數學發展的影響也至關重要；書中包括的課題有：平行移動及其應用、地圖設色、完整的高斯麯率。

目錄

Preface
Contents
1．Curves in the plane and in space
1．1 What is a curve？
1．2 Arc-length
1．3 Reparametrization
1．4 Closed curves
1．5 Level curves versus parametrized curves

2．How much does a curve curve？
2．1 Curvature
2．2 Plane curves
2．3 Space curves

3．Global properties of curves
3．1 Simple closed curves
3．2 The isoperimetric inequality
3．3 The four vertex theorem

4．Surfaces in three dimensions
4．1 What is a surface？
4．2 Smooth surfaces
4．3 Smooth maps
4．4 Tangents and derivatives
4．5 Normals and orientability

5．Examples of surfaces
5．1 Level surfaces
5．2 Quadric surfaces
5．3 Ruled surfaces and surfaces of revolution
5．4 Compact surfaces
5．5 Triply orthogonal systems
5．6 Applications of the inverse function theorem

6．The flrst fundamental form
6．1 Lengths of curves on surfaces
6．2 Isometries of surfaces
6．3 Conformal mappings of surfaces
6．4 Equiareal maps and a theorem of Archimedes
6．5 Sphericalgeometry

7．Curvature of 8urfaces
7．1 The second fundamental form
7．2 The Gauss and Weingarten maps
7．3 Normal and geodesic curvatures
7．4 Parallel transport and covariant derivative

8．Gaussian， mean and principal curvatures
8．1 Gaussian and mean curvatures
8．2 Principal curvatures of a surface
8．3 Surfaces of constant Gaussian curvature
8．4 Flat surfaces
8．5 Surfaces of constant mean curvature
8．6 Gaussian curvature of compact surfaces

9．Geodesics
9．1 Definition and basic properties
9．2 Geodesic equations
9．3 Geodesics on surfaces of revolution
9．4 Geodesics as shortest paths
9．5 Geodesic coordinates

10．Gauss' Theorema Egregium
10．1 The Gauss and Codazzi-Mainardi equations
10．2 Gauss' remarkable theorem
10．3 Surfaces of constant Gaussian curvature
10．4 Geodesic mappings

11．Hyperbolic geometry
11．1 Upper half-plane model
11．2 Isometries of H
11．3 Poincare disc model
11．4 Hyperbolic parallels
11．5 Beltrami-Klein model

12．Minmal surfaces
12．1 Plateau's problem
12．2 Examples of minimal surfaces
12．3 Gauss map of a minimal surface
12．4 Conformal parametrization of minimal surfaces
12．5 Minimal surfaces and holomorphic functions

13．The Gauss-Bonnet theorem
13．1 Gauss-Bonnet for simple closed curves
13．2 Gauss-Bonnet for curvilinear polygons
13．3 Integration on compact surfaces
13．4 Gauss-Bonnet for compact surfaces
13．5 Map colouring
13．6 Holonomy and Gaussian curvature
13．7 Singularities of vector fields
13．8 Critical points

A0．Inner product spaces and self-adjoint linear maps
A1．Isometries of Euclidean spaces
A2．Mobius transformations
Hints to selected exercises
Solutions
Index
微分幾何基礎（第2版 英文版） [Elementary Differential Ceometry （Second Edition）] 下載 mobi epub pdf txt 電子書

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