分析流形和物理学（第1卷·基础·修订版）

图书标准信息

• 作者 Yvonne、Cecile Dewitt-Morette 编
• 出版社 世界图书出版公司
• 出版时间 2010-09
• 版次 1
• ISBN 9787510027284
• 定价 69.00元
• 装帧 平装
• 开本 32开
• 纸张 胶版纸
• 页数 630页
• 正文语种 英语

【内容简介】

Alltooofteninphysicsfamiliarityisasubstituteforunderstanding，andthebeginnerwholacksfamiliaritywonderswhichisatfault：physicsorhimself.Physicalmathematicsprovideswelldefinedconceptsandtechni-quesforthestudyofphysicalsystems.Itismorethanmathematicaltechniquesusedinthesolutionofproblemswhichhavealreadybeenformulated;ithelpsintheveryformulationofthelawsofphysicalsystemsandbringsabetterunderstandingofphysics.Thusphysicalmathematicsincludesmathematicswhichgivespromiseofbeingusefulinouranalysisofphysicalphenomena.Attemptstousemathematicsforthispurposemayfailbecausethemathematicaltoolistoocrude;physicsmaythenindicatealongwhichlinesitshouldbesharpened.Infact，theanalysisofphysicalsystemshasspurredmanyanewmathematicaldevelopment.
Considerationsofrelevancetophysicsunderliethechoiceofmaterialincludedhere.Anychoiceisnecessarilyarbitrary;weincludedfirstthetopicswhichweenjoymostbutwesoonrecognizedthatinstantgratifica-tionisashortrangecriterion.Wethenincludedmaterialwhichcanbeappreciatedonlyafteragreatdealofintellectualasceticismbutwhichmaybefartherreaching.Finally，thisbookgathersthestartingpointsofsomegreatcurrentsofcontemporarymathematics.Itisintendedforanadvancedphysicalmathematicscourse.

【目录】

i.reviewoffundamentalnotionsofanalysis
a.settheory，definitions
1.sets
2.mappings
3.relations
4.orderings
b.algebraicstructures，definitions
1.groups
2.rings
3.modules
4.algebras
5.linearspaces
c.topology
1.definitions
2.separation
3.base
4.convergence
5.coveringandcompactness
6.connectedness
7.continuousmappings
8.multipleconnectedness
9.associatedtopologies
10.topologyrelatedtootherstructures
11.metricspaces
metricspaces
cauchysequence;completeness
12.banachspaces
normedvectorspaces
banachspaces
strongandweaktopology;compactedness
13.hilbertspaces
d.integration
1.introduction
2.measures
3.measurespaces
4.measurablefunctions
5.lntegrablefunctions
6.integrationonlocallycompactspaces
7.signedandcomplexmeasures
8.integrationofvectorvaluedfunctions
9.l1space
10.l1space
e.keytheoremsinlinearfunctionalanalysis
1.boundedlinearoperators
2.compactoperators
3.openmappingandclosedgraphtheorems
problemsandexercises
problem1：cliffordalgebra;spin（4）
exercise2：producttopology
problem3：strongandweaktopologiesinl2
exercise4：htlderspaces
seeproblemvi4：applicationtotheschrtdingerequation
ii.differentialcalculusonbanachspaces
a.foundations
1.definitions.taylorexpansion
2.theorems
3.diffeomorphisms
4.theeulerequation
5.themeanvaluetheorem
6.higherorderdifferentials
b.calculusofvariations
1.necessaryconditionsforminima
2.sufficientconditions
3.lagrangianproblems
c.implicitfunctiontheorem.inversefunctiontheorem
1.contractingmappingtheorems
2.inversefunctiontheorem
3.implicitfunctiontheorem
4.globaltheorems
d.differentialequations
1.firstorderdifferentialequation
2.existenceanduniquenesstheoremsforthelipschitziancase
problemsandexercises
problem1：banachspaces，firstvariation，linearizedequation
problem2：taylorexpansionoftheaction;jacobifields;thefeynman-greenfunction;thevanvleckmatrix;conjugatepoints;caustics
problem3：euler-lagrangeequation;thesmalldisturbanceequation;thesoapbubbleproblem;jacobifields
iii.differentiablemanifolds，finitedimensionalcase
a.definitions
1.differentiablemanifolds
2.diffeomorphisms
3.liegroups
b.vectorfields;tensorfields
1.tangentvectorspaceatapoint
tangentvectorasaderivation
tangentvectordefinedbytransformationproperties
tangentvectorasanequivalenceclassofcurves
imagesunderdifferentiablemappings
2.fibrebundles
definition
bundlemorphisms
tangentbundle
framebundle
principalfibrebundle
3.vectorfields
vectorfields
movingframes
imagesundercliffeomorphisms
4.covariantvectors;cotangentbundles
dualofthetangentspace
spaceofdifferentials
cotangentbundle
reciprocalimages
5.tensorsatapoint
tensorsatapoint
tensoralgebra
6.tensorbundles;tensorfields
c.groupsoftransformations
i.vectorfieldsasgeneratorsoftransformationgroups
2.liederivatives
3.invarianttensorfields
d.liegroups
1.definitions;notations
2.leftandrighttranslations;liealgebra;structureconstants
3.one-parametersubgroups
4.exponentialmapping;taylorexpansion;canonicalcoordinates
5.liegroupsoftransformations;realization
6.adjointrepresentation
7.canonicalform，maurer--cartanform
problemsandexercises
problem1：changeofcoordinatesonafiberbundle，configurationspace，phasespace
problem2：liealgebrasofliegroups
problem3：thestraintensor
problem4：exponentialmap;taylorexpansion;adjointmap;leftandrightdifferentials;haarmeasure
problem5：thegroupmanifoldsofsoo）andsu（2）
problem6：the2-sphere
iv.integrationonmanifolds
a.exteriordifferentialforms
1.exterioralgebra
exteriorproduct
localcoordinates;strictcomponents
changeofbasis
2.exteriordifferentiation
3.reciprocalimageofaform（pullback）
4.derivationsandantiderivations
definitions
interiorproduct
5.formsdefinedonaliegroup
invariantforms
maurer--cartanstructureequations
6.vectorvalueddifferentialforms
b.integration
1.integration
orientation
oddforms
integrationofn-formsinr"
partitionsofunity
propertiesofintegrals
2.stokestheorem
p-chains
integralsofp-formsonp-chains
boundaries
mappingsofchains
proofofstokestheorem
3.globalproperties
homologyandcohomology
o-formsando-chains
bettinumbers
poincar6lemmas
derhamandpoincaredualitytheorems
cexteriordifferentialsystems
1.exteriorequations
2.singleexteriorequation
3.systemsofexteriorequations
idealgeneratedbyasystemofexteriorequations
algebraicequivalence
solutions
examples
4.exteriordifferentialequations
integralmanifolds
associatedpfaffsystems
genericpoints
closure
5.mappingsofmanifolds
introduction
immersion
embedding
submersion
6.pfaffsystems
completeintegrability
frobeniustheorem
integrabilitycriterion
examples
dualformofthefrobeniustheorem
7.characteristicsystem
characteristicmanifold
example：firstorderpartialdifferentialequations
completeintegrability
constructionofintegralmanifolds
cauchyproblem
examples
8.invariants
invariantwithrespecttoapfaffsystem
integralinvariants
9.example：integralinvariantsofclassicaldynamics
liouvilletheorem
canonicaltransformations
10.symplecticstructuresandhamiltoniansystems
problemsandexercises
problem1：compoundmatrices
problem2：poincar6lemma，maxwellequations，wormholes
problem3：integralmanifolds
problem4：firstorderpartialdifferentialequations，hamilton-jacobi
equations，lagrangianmanifolds
problem5：firstorderpartialdifferentialequations，catastrophes
problem6：darbouxtheorem
problem7：timedependenthamiltonians
seeproblemvi11paragraphc：electromagneticshockwaves
v.riemannianmanifolds.kahlerianmanifolds
a.theriemannianstructure
1.preliminaries
metrictensor
hyperbolicmanifold
2.geometryofsubmanifolds，inducedmetric
3.existenceofariemannianstructure
properstructure
hyperbolicstructure
euler-poincarecharacteristic
4.volumeelement.thestaroperator
volumeelement
staroperator
5.isometries
b.linearconnections
1.linearconnections
covariantderivative
connectionforms
paralleltranslation
affinegeodesic
torsionandcurvature
2.riemannianconnection
definitions
locallyflatmanifolds
3.secondfundamentalform
4.differentialoperators
exteriorderivative
operator
divergence
laplacian
c.geodesics
1.arclength
2.variations
eulerequations
energyintegral
3.exponentialmapping
definition
normalcoordinates
4.geodesicsonaproperriemannianmanifold
properties
geodesiccompleteness
5.geodesicsonahyperbolicmanifold
d.almostcomplexandkahlerianmanifolds
problemsandexercises
problem1maxwellequation;gravitationalradiation
problem2：theschwarzschildsolution
problem3：geodeticmotion;equationofgeodeticdeviation;exponentiation;conjugatepoints
problem4：causalstructures;conformalspaces;weyltensor
vbis.connectionsonaprincipalfibrebundle
a.connectionsonaprincipalfibrebundle
1.definitions
2.localconnectionl-formsonthebasemanifold
existencetheorems
sectioncanonicallyassociatedwithatrivialization
potentials
changeoftrivialization
examples
3.covariantderivative
associatedbundles
paralleltransport
covariantderivative
examples
4.curvature
definitions
cartanstructuralequation
localcurvatureonthebasemanifold
fieldstrength
bianchiidentities
5.linearconnections
definition
solderingform，torsionform
torsionstructuralequation
standardhorizontal（basic）vectorfield
curvatureandtorsiononthebasemanifold
bundlehomomorphism
metricconnection
b.hoionomy
1.reduction
2.holonomygroups
c.characteristicclassesandinvariantcurvatureintegrals
1.characteristicclasses
2.gauss-bonnettheoremandchernnumbers
3.theatiyah-singerindextheorem
problemsandexercises
problem1：thegeometryofgaugefields
problem2：chargequantization.monopoles
problem3：instantonsolutionofeuclideansu（2）yang-millstheory（connectiononanon-trivialsu（2）bundleovers4）
problem4：spinstructure;spinors;spinconnections
vi.distributions
a.testfunctions
1.seminorms
definitions
hahn-banachtheorem
topologydefinedbyafamilyofseminorms
2.d-spaces
definitions
inductivelimittopology
convergenceindm（u）andd（u）
examplesoffunctionsin
truncatingsequences
densitytheorem
b.distributions
1.definitions
distributions
measures;diracmeasuresandlerayforms
distributionoforderp
supportofadistribution
distributionswithcompactsupport
2.operationsondistributions
sum
productbycfunction
directproduct
derivations
examples
inversederivative
3.topologyond
weakstartopology
criterionofconvergence
4.changeofvariablesinrn
changeofvariablesinrn
transformationofadistributionunderadiffeomorphism
invariance
5.convolution
convolutionalgebral1（rn）
convolutionalgebrad+andd-
derivationandtranslationofaconvolutionproduct
regularization
supportofaconvolution
equationsofconvolution
differentialequationwithconstantcoefficients
systemsofconvolutionequations
kernels
6.fouriertransform
fouriertransformofintegrablefunctions
tempereddistributions
fouriertransformoftempereddistributions
paley-wienertheorem
fouriertransformofaconvolution
7.distributiononac∞paracompactmanifold
8.tensordistributions
c.sobolevspacesandpartialdifferentialequations
i.sobolevspaces
properties
densitytheorems
w?spaces
fouriertransform
planchereltheorem
sobolevsinequalities
2.partialdifferentialequations
definitions
cauchy-kovalevskitheorem
classifications
3.ellipticequations;laplacians
elementarysolutionoflaplacesequation
subharmonicdistributions
potentials
energyintegral，greensformula，unicitytheorem
liouvillestheorem
boundary-valueproblems
greenfunction
introductiontohilbertianmethods;generalizeddirichletproblem
hilbertianmethods
example：neumannproblem
4.parabolicequations
heatdiffusion
5.hyperbolicequation;waveequations
elementarysolutionofthewaveequation
cauchyproblem
energyintegral，unicitytheorem
existencetheorem
6.leraytheoryofhyperbolicsystems
7.secondordersystems;propagators
problemsandexercises
problem1：boundeddistributions
problem2：laplacianofadiscontinuousfunction
exercise3：regularizedfunctions
problem4：applicationtotheschrbdingerequation
exercise5：convolutionandlinearcontinuousresponses
problem6：fouriertransformsofexp（-x2）andexp（ix2）
problem7：fouriertransformsofheavisidefunctionsandpr（l/x）
problem8：diracbitensors
problem9：legendrecondition
problem10：hyperbolicequations;characteristics
problem11：electromagneticshockwaves
problem12：elementarysolutionofthewaveequation
problem13：elementarykernelsoftheharmonicoscillator
vii.differentiablemanifolds，infinitedimensionalcase
a.infinite-dimensionalmanifolds
1.definitionsandgeneralproperties
e-manifolds
differentiablefunctions
tangentvector
vectorandtensorfield
differentialofamapping
submanifold
immersion，embedding，submersion
flowofavectorfield
differentialforms
2.symplecticstructuresandhamiltoniansystems
definitions
complexstructures
canonicalsymplecticform
symplectictransformation
hamiltonianvectorfield
conservationofenergytheorem
riemannianmanifolds
b.theoryofdegree;leray-schaudertheory
i.definitionforfinitedimensionalmanifolds
degree
integralformulaforthedegreeofafunction
continuousmappings
2.propertiesandapplications
fundamentaltheorem
borsukstheorem
brouwersfixedpointtheorem
producttheorem
3.leray-schaudertheory
definitions
compactmappings
degreeofacompactmapping
schauderfixedpointtheorem
leray-schaudertheorem
c.morsetheory
1.introduction
2.definitionsandtheorems
3.indexofacriticalpoint
4.criticalnecktheorem
d.cylindricalmeasures，wienerintegral
1.introduction
2.promeasuresandmeasuresonalocallyconvexspace
projectivesystem
promeasures
imageofapromeasure
integrationwithrespecttoapromeasureofacylindrical
function
fouriertransforms
3.gaussianpromeasures
gaussianmeasuresonrn
gaussianpromeasures
gaussianpromeasuresonhilbertspaces
4.thewienermeasure
wienerintegral
sequentialwienerintegral
problemsandexercises
problema：theklein-gordonequation
problemb：applicationoftheleray-schaudertheorem
problemc1：thereebtheorem
problemc2：themethodofstationaryphase
problemd1：ametriconthespaceofpathswithfixedendpoints
problemd2：measuresinvariantundertranslation
problemd3：cylindricalσ-fieldofc（[a，b]）
problemd4：generalizedwienerintegralofacylindricalfunction
references
symbols
index