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现代几何学方法和应用（英文版）

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作者：B.A.Dubrovin，A.T.Fomenko,et al. 编著

出版社：世界图书出版公司

出版时间：1999-11-1

I S B N：9787506201230

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现代几何学方法和应用-(第1卷)(第2版)

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现代几何学方法和应用（英文版）

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内容简介

Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) gradually came to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say,which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstness of their exposition The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstness of its exposition were dictated by the view that, in addition to the geometry,of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity,to name but two), and are therefore essential: the theory of tensors (including eovariant differentiation of them); Riemannian curvature;

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Preface to the First Edition
CHAPTER 1 Geometry in Regions of a Space. Basic Concepts
1. Co-ordinate systems
1.1. Cartesian co-ordinates in a space
1.2. Co-ordinate changes
2. Euclidean space
2.1. Curves in Euclidean space
2.2. Quadratic forms and vectors
3. Riemannian and pseudo-Riemannian spaces
3.1. Riemannian metrics
3.2. The Minkowski metric
4. The simplest groups of transformations of Euclidean space
4.1. Groups of transformations of a region
4.2. Transformations of the plane
4.3. The isometries of 3-dimensional Euclidean space
4.4. Further examples of transformation groups
4.5. Exercises
5. The Serret-Frenet formulae
5.1.Curvature of curves in the Euclidean plane
5.2.Cuuves in Euclidean 3-space.Curvature and torsion
5.3.Orthogonal transformations depending on a parameter
5.4.Exercises
6.Pseudo-Euclidean spaces
6.1.The simplest concepts of the special theory of relativity
6.2.Lorentz transformations
6.3.Excercises
CHAPTER 2 The Theory of Surfaces
……
CHAPTER 3 Tensors:The Algebrai Theory
CHAPTER 4 The Differential Calculus of Tensors
CHAPTER 5 The Elements of the Calclus of Variations
CHAPTER 6 The Calculus of Variations in Several Dimensions.
Bibliography
Index