物理學(xué)家用的微分幾何和李群

前言

　　This is an introductory text dealing with a part of mathematics： modem differential geometry and the theory of Lie groups. It is written from the perspective of and mainly for the needs of physicists. The orientation on physics makes itself felt in the choice of material， in the way it is presented （e.g. with no use of a definition-theorem-proof scheme）， as well as in the content of exercises （often they are closely related to physics）.　　Its potential readership does not， however， consist of physicists alone. Since the book is about mathematics， and since physics has served for a fairly long time as a rich source of inspiration for mathematics， it might be useful for the mathematical community as well. More generally， it is suitable for anybody who has some （rather modest） preliminary background knowledge （to be specified in a while） and who desires to become familiar in a comprehensible way with this interesting， important and living subject， which penetrates increasingly into various branches of modem theoretical physics， "pure" mathematics itself， as well as into its numerous applications.

內(nèi)容概要

　微分幾何在現(xiàn)代理論物理和應(yīng)用數(shù)學(xué)中扮演著越來越重要的角色。本書給出了在理論物理和應(yīng)用數(shù)學(xué)中很重要的幾何知識的引入，包括，流形、張量場、微分形式、聯(lián)絡(luò)、辛幾何、李群作用、族以及自旋?！”緯砸环N非正式的形式寫作，作者給出了1000多例子重在強(qiáng)調(diào)對一般理論的深刻理解。本書將要為讀者很好的學(xué)習(xí)拉格郎日現(xiàn)代處理方法、哈密頓力學(xué)、電磁、規(guī)范場，相對論以及萬有引力做充足的準(zhǔn)備。　本書很適合作為物理、數(shù)學(xué)以及工程專業(yè)的高年級本科生以及研究生的教程，也是一本很難得自學(xué)教程。　

書籍目錄

PrefaceIntroduction1　The　concept　of　a　manifold2　Vector　and　tensor　fields3　Mappings　of　tensors　induced　by　mappings　of　manifolds4　Lie　derivative5　Exterior　algebra6　Differential　calulus　of　forms7　Integral　calculus　of　forms8　Particular　cases　and　applications　of　Stokes'theorem9　Poincare　lemma　and　cohomologies10　Lie　groups:basic　facts11　Differential　geometry　on　Lie　groups12　Representations　of　Lie　groups　and　Lie　algebras13　Actions　of　Lie　groups　and　Lie　algebras　on　manifolds14　Hamiltonian　mechanics　and　symplectic　manifolds15　Parallel　transport　and　linear　connection　on　M16　Field　theory　and　the　language　of　forms17　Differential　geometry　on　T　M　and　T*M18　Hamiltonian　and　Lagrangian　equations19　Linear　connection　and　the　frame　bundle20　Connection　on　a　principal　G-bundle21　Gauge　theories　and　connections22　Spinor　fields　and　the　Dirac　operatorAppendix　A　Some　relevant　algebrai　structuresAppendix　B　Starring

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