数学物理的几何方法

2009-6  

世界图书出版公司  

舒茨  

250  

无  

Why study geometry？This book alms to introduce the beginning or working physicist to awide range of aualytic tools which have their or/gin in differential geometry andwhich have recently found increasing use in theoretical physics. It is not uncom-mon today for a physicist's mathematical education to ignore all but the sim-plest geometrical ideas, despite the fact that young physicists are encouraged todevelop mental 'pictures' and 'intuition' appropriate to physical phenomena.This curious neglect of 'pictures' of one's mathematical tools may be seen as the outcome of a gradual evolution over many centuries. Geometry was certainly extremely important to ancient and medieval natural philosophers; it was ingeometrical terms that Ptolemy, Copernicus, Kepler, and Galileo all expressedtheir thinking. But when Descartes introduced coordinates into Euclideangeometry, he showed that the study of geometry could be regarded as an appli.cation of algrebra. Since then, the/mportance of the study of geometry in theeducation of scientists has steadily declined, so that at present a university undergraduate physicist or applied mathematician is not likely to encounter much geometry at all. One reason for this suggests itself immediately: the relatively simple geometry of the three-dimensional Euclidean world that the nineteenth-century physicist believed he lived in can be mastered quickly, while learning the great diversity ofanalytic techniques that must be used to solve the differential equations of physics makes very heavy demands on the student's time. Another reason mustsurely be that these analytic techniques were developed at least partly in response to the profound realization by physicists that the laws of nature couldbe expressed as differential equations, and th/s led most mathematical physicists genuinely to neglect geometry until relatively recently. However, two developments in this century have markedly altered the balancebetween geometry and analysis in the twentieth-century physicist's outloook.The first is the development of the theory of relativity, according to which theEuclidean three-space of the nineteenth-century physicist is only an approximation to the correct description of the physical world. The second development,which is only beginning to have an impact.

　　This book alms to introduce the beginning or working physicist to awide range of aualytic tools which have their or/gin in differential geometry andwhich have recently found increasing use in theoretical physics. It is not uncom-mon today for a physicists mathematical education to ignore all but the sim-plest geometrical ideas， despite the fact that young physicists are encouraged todevelop mental pictures and intuition appropriate to physical phenomena.This curious neglect of pictures of ones mathematical tools may be seen as the outcome of a gradual evolution over many centuries. Geometry was certainly extremely important to ancient and medieval natural philosophers; it was ingeometrical terms that Ptolemy， Copernicus， Kepler， and Galileo all expressedtheir thinking. But when Descartes introduced coordinates into Euclideangeometry， he showed that the study of geometry could be regarded as an appli.cation of algrebra. Since then， the/mportance of the study of geometry in theeducation of scientists has steadily

作者：(英国)舒茨(Schutz.B.)

1 Some basic mathematics　1.1 The space Rn and its topology　1.2 Mappings　1.3 Real analysis　1.4 Group theory　1.5 Linear algebra　1.6 The algebra of square matrices　1.7 Bibliography2 Dffferentiable manifolds and tensors　2.1 Def'mition of a manifold　2.2 The sphere as a manifold　2.3 Other examples of manifolds　2.4 Global considerations　2.5 Curves　2.6 Functions on M　2.7 Vectors and vector fields　2.8 Basis vectors and basis vector fields　2.9 Fiber bundles　2.10 Examples of fiber bundles　2.11 A deeper look at fiber bundles　2.12 Vector fields and integral curves　2.13 Exponentiation of the operator d/dZ　2.14 Lie brackets and noncoordinate bases　2.15 When is a basis a coordinate basis？　2.16 One-forms　2.17 Examples of one-forms　2.18 The Dirac delta function　2.19 The gradient and the pictorial representation of a one-form　2.20 Basis one-forms and components of one-forms　2.21 Index notation　2.22 Tensors and tensor fields　2.23 Examples of tensors　2.24 Components of tensors and the outer product　2.25 Contraction　2.26 Basis transformations　2.27 Tensor operations on components　2.28 Functions and scalars　2.29 The metric tensor on a vector space　2.30 The metric tensor field on a manifold　2.31 Special relativity　2.32 Bibliography3 Lie derivatives and Lie groups　3.1 Introduction: how a vector field maps a manifold into itself　3.2 Lie dragging a function　3.3 Lie dragging a vector field　3.4 Lie derivatives　3.5 Lie derivative of a one-form　3.6 Submanifolds　3.7 Frobenius' theorem (vector field version)　3.8 Proof of Frobenius' theorem　3.9 An example: the generators ors2　3.10 Invariance　3.11 Killing vector fields　3.12 Killing vectors and conserved quantities in particle dynamics　3.13 Axial symmetry　3.14 Abstract Lie groups　3.15 Examples of Lie groups　3.16 Lie algebras and their groups　3.17 Realizations and representatidns　3.18 Spherical symmetry, spherical harmonics and representations of the rotation group　3.19 Bibliography4 Differential forms A The algebra and integral calculus of forms　4.1 Definition of volume - the geometrical role of differential forms　4.2 Notation and definitions for antisymmetric tensors　4.3 Differential forms　4.4 Manipulating differential forms　4.5 Restriction of forms　4.6 Fields of forms5 Applications in physics　A Thermodynamics6 Connections for Riemannian manifolds and gauge theoriesAppendix: solutions and hints for selected exercisesNotationIndex

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《数学物理的几何方法(英文版)》是由世界图书出版公司出版的。

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讲述了分析与几何的关系，作者希望能培养读者的物理直觉思维以及图像思维的能力。

不是太清楚此书如何，但是当当的送货速度以及态度是相当的好，呵呵

总体很好，质量不错，内容很有用，价格也合理。但如果有中文版就更方便了。

很旧，书页都发黄了，真的不怎么样。。

这本书是老师推荐的，比较基础，适合学物理的人看。现在偏向于理论物理的同学都应该知道一些广义相对论，而理解广义相对论的前提就是要掌握一些微分几何的东西。另外，不要抱着只看一本书就能把微分几何弄清楚的心态，需肯定要对比其他书来增进理解，比如一开始的拓扑，对于初学者就很难理解，而对于讲微分几何的书来说又不可能讲的特别多，就像梁灿彬老师的课一样，拓扑也只是讲了两课时。这应该是影印版，肯定不如原版的好，但还不至于影响阅读，值得购买。估计看到评论的人很少，因为需要这种书的人太少了

是喷墨打印机打的吧，手上有点汗了，一不小心字就模糊了，可惜这本好书了。

如题，但书中内容程度不一，看似比较适合表达全面的内容，却很少顾及实际读者自学的效果。由于本书较老，可以看图书馆的中译版，下个电子版比对也很好。