Modern geometry-methods and applications 9787519296698 B. A.

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書名：Modern geometry-methods and applications
ISBN：9787519296698
出版社：世界圖書出版公司
著編譯者：B. A. Dubrovin A. T. Fomenko S
頁數：468
所在地：中國大陸 *此為代購商品
書號：1513302
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【台灣高等教育出版社簡體書】 Modern geometry-methods and applications 787519296698 B. A. Dubrovin A. T. Fomenko S

內容簡介

本書是莫斯科大學數學力學系經典教材《現代幾何學——方法和應用》三卷本的第1卷。全書力求以直觀的和物理的視角闡述，是一本難得的現代幾何方面的佳作。整套書內容包括張量分析、曲線和曲面幾何、一維和高維變分法（第1卷），微分流形的拓撲和幾何（2卷），以及同調與上同調理論（第3卷）。本書可用作數學和理論物理專業高年級和研究生的教學用書，對從事幾何和拓撲研究的工作者也極具參考價值。

目錄

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 Curves 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 Exercises
CHAPTER 2 The Theory of Surfaces
7 Geometry on a surface in space
7 1 Co-ordinates on a surface
7 2 Tangent planes
7 3 The metric on a surface in Euclidean space
7 4 Surface area
7 5 Exercises
8 The second fundamental form
8 1 Curvature of curves on a surface in Euclidean space
8 2 Invariants of a pair of quadratic forms
8 3 Properties of the second fundamental form
8 4 Exercises
9 The metric on the sphere
10 Space-like surfaces in pseudo-Euclidean space
10 1 The pseudo-sphere
10 2 Curvature of space-like curves in R
11 The language of complex numbers in geometry
11 1 Complex and real co-ordinates
11 2 The Hermitian scalar product
11 3 Examples of complex transformation groups
12 Analytic functions
12 1 Complex notation for the element of length， and for the differential of a function
12 2 Complex co-ordinate changes
12 3 Surfaces in complex space
13 The conformal form of the metric on a surface
13 1 Isothermal co-ordinates Gaussian curvature in terms of conformal co-ordinates
13 2 Conformal form of the metrics on the sphere and the Lobachevskian plane
13 3 Surfaces of constant curvature
13 4 Exercises
14 Transformation groups as surfaces in N-dimensional space
14 1 Co-ordinates in a neighbourhood of the identity
14 2 The exponential function with matrix argument
14 3 The quaternions
14 4 Exercises
15 Conformal transformations of Euclidean and pseudo-Euclidean spaces of several dimensions
CHAPTER 3 Tensors: The Algebraic Theory
16 Examples of tensors
17 The general definition of a tensor
17 1 The transformation rule for the components of a tensor of arbitrary rank
17 2 Algebraic operations on tensors
17 3 Exercises
18 Tensors of type (0，k)
18 1 Differential notation for tensors with lower indices only
18 2 Skew-symmetric tensors of type (0， k)
18 3 The exterior product of differential forms The exterior algebra
18 4 Skew-symmetric tensors of type (k， 0)(polyvectors) Integrals with respect to anti-commuting variables
18 5 Exercises
19 Tensors in Riemannian and pseudo-Riemannian spaces
19 1 Raising and lowering indices
19 2 The eigenvalues of a quadratic form
19 3 The operator
19 4 Tensors in Euclidean space
19 5 Exercises
20 The crystallographic groups and the finite subgroups of the rotation group of Euclidean 3-space Examples of invariant tensors
21 Rank 2 tensors in pseudo-Euclidean space， and their eigenvalues
21 1 Skew-symmetric tensors The invariants of an electromagnetic field
21 2 Symmetric tensors and their eigenvalues The energy-momentum tensor of an electromagnetic feltauvs v
22 The behaviour of tensors under mappings
22 1 The general operation of restriction of tensors with lower indices
22 2
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