微分幾何基礎(chǔ)

前言

　　This book is an elementary account of the geometry of curves and surfaces. It is written for students who have completed standard courses in calculus and linear algebra， and its aim is to introduce some of the main ideas of differential geometry.　　The language of the book is established in Chapter 1 by a review of the core content of differential calculus， emphasizing linearity. Chapter 2 describes the method of moving frames， which is introduced， as in elementary calculus， to study curves in space. （This method turns out to apply with equal efficiency to surfaces.） Chapter 3 investigates the rigid motions of space， in terms of which congruence of curves and surfaces is defined in the same way as congruence of triangles in the plane.　　Chapter 4 requires special comment. One weakness of classical differential geometry is its lack of any adequate definition of surface. In this chapter we decide just what a surface is， and show that every surface has a differential and integral calculus of its own， strictly analogous to the familiar calculus of the plane. This exposition provides an introduction to the notion of differentiable manifold， which is the foundation for those branches of mathematics and its applications that are based on the calculus.　　The next two chapters are devoted to the geometry of surfaces in 3space. Chapter 5 measures the shape of a surface and derives basic geometric invariants， notably Gaussian curvature. Intuitive and computational aspects are stressed to give geometrical meaning to the theory in Chapter 6.

內(nèi)容概要

本書介紹曲線和曲面幾何的入門知識(shí)，主要內(nèi)容包括歐氏空間上的積分、幀場(chǎng)、歐氏幾何、曲面積分、形狀算子、曲面幾何、黎曼幾何、曲面上的球面結(jié)構(gòu)等。修訂版擴(kuò)展了一些主題，更加強(qiáng)調(diào)拓?fù)湫再|(zhì)、測(cè)地線的性質(zhì)、向量場(chǎng)的奇異性等。更為重要的是，修訂版增加了計(jì)算機(jī)建模的內(nèi)容，提供了Mathematica和Maple程序。此外，還增加了相應(yīng)的計(jì)算機(jī)習(xí)題，補(bǔ)充了奇數(shù)號(hào)碼習(xí)題的答案，更便于教學(xué)?！　”緯m合作為高等院校本科生相關(guān)課程的教材，也適合作為相關(guān)專業(yè)研究生和科研人員的參考書。

作者簡(jiǎn)介

　　Barrett ONeill，加州大學(xué)洛杉磯分校教授。1951年在麻省理丁學(xué)院獲得博士學(xué)位。他的研究方向包括：曲線和曲面幾何，計(jì)算機(jī)和曲面，黎曼幾何，黑洞理論等。另著有Semi-Riemannian Geometry with Applications to Relativity和The Geometry of Kerr Black Holes等書。

書籍目錄

1. Calculus on Euclidean Space　1.1. Euclidean Space　　1.2. Tangent Vectors　　1.3. Directional Derivatives　　1.4. Curves in R3　　1.5. 1-Forms　　1.6. Differential Forms　　1.7. Mappings　　1.8. Summary　2. Frame Fields 　2.1.Dot Product　　2.2. Curves　　2.3. The Frenet Formulas　　2.4. Arbitrary-speed Curves　　2.5. Covariant Derivatives　 　2.6. Frame Fields　　2.7. Connection Forms　　2.8. The Structural Equations　3.Euclidean Geometry　3.1.Isometries of R3　3.2.The Tangent Map of an Isometry　3.3.Orientation　3.4.Euclidean Geometry　3.5.Congruence of Curves　3.6.Summary4.Calculus on a SUrface　4.1.Surfaces in R3　4.2.Patch Computations　4.3.Differentiable Functions and Tangent Vectors　4.4.Differential Forms on a Surface　4.5.Mappings of Surfaces　4.6.Integration of Forms　4.7.Topological Properties of Surfaces　4.8.Manifcllds　4.9.Summary5.Shape Operators　5.1.The Shape Operator of M c R3　5.2.Normal Curvature　5.3.Gaussian Curvature　5.4.Computational Techniques　5.5.The Implicit Case　5.6.Special Curves in a Surface　5.7.Surfaces of Revolution　5.8.Summary6.Geometry Of Sudaces in R3　6.1.The Fundamental Equations　6.2.Form Computations　6.3.Some Global Theorems　6.4.Isometries and Local Isometries　6.5.Intrinsic Geometry of Surfaces in R3　6.6.Orthogonal Coordinates　6.7.Integration and Orientation　6.8.Total Curvature　6.9.Congruence of Surfaces　6.10.Summary7.Riemannian Geometry　7.1.Geometric Surfaces　7.2，Gaussian Curvature　7.3.Covariant Derivative　7.4.Geodesics　7.5.Clairaut Parametrizations　7.6.The Gauss.Bonnet Theorem　7.7.Applications of Gauss。Bonnet　7.8.Summary8.GIobaI Structure of Suffaces　8.1.Length.Minimizing Properties of Geodesics　8.2.Complete Surfaces　8.3，Curvature and Conjugate Points　8.4.Covering Surfaces　8.5.Mappings That Preserve Inner Products　8.6.Surfaces of Constant Curvature　8.7.Theorems of Bonnet and Hadamard　8.8.SummaryAppendix：Computer FormulasBibliographyAnswers to Odd-Numbered Exerciseslndex

章節(jié)摘錄

　　As mentioned in the Preface，the purpose of this initial chapter is to establish the mathematical language used throughout the book.Much of what we do is simply a review of that part of elementary calculus dealing with differentiation of functions of three variables and with curves in space.Our deftnitions have been formulated SO that they will apply smoothly to the later study of surfaces.　　1.1 Euclidean Space　　Three-dimensional space is often used in mathematics without being formally defined.Looking at the corner of a room，one can picture the familiar process by which rectangular coordinate axes are introduced and three numbers are measured to describe the position of each point.A precise definition that realizes this intuitive picture may be obtained by this device：instead of saying that three numbers describe the position of a point，we define them to be a point.

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