All the Math You Missed (But Need to Know for Graduate School), Second Edition [2nd Ed]

417 Pages·2021·1.949 MB·English

by  Thomas A. Garrity

All the Math You Missed
Beginning graduate students in mathematical sciences and related areas in physical and
computersciencesandengineeringareexpectedtobefamiliarwithadauntingbreadthof
mathematics,butfewhavesuchabackground.Thisbestsellingbookhelpsstudentsfillin
thegapsintheirknowledge.
ThomasA.Garrityexplainsthebasicpointsandafewkeyresultsofallthemostimportant
undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The
explanationsareaccompaniedbynumerousexamples,exercisesandsuggestionsforfurther
readingthatallowthereadertotestanddeveloptheirunderstandingofthesecoretopics.
Featuring four new chapters and many other improvements, this second edition of All
theMathYouMissed isanessentialresourceforadvancedundergraduatesandbeginning
graduatestudentswhoneedtolearnsomeseriousmathematicsquickly.
Thomas A. Garrity is the Webster Atwell Class of 1921 Professor of Mathematics at
Williams College, Massachusetts, where he was the director of the Williams College
Project for Effective Teaching for many years. Among his awards are Rice University’s
NicolasSalgoOutstandingTeachingawardandtheHaimoawardoftheMAA.Hisother
booksincludeAlgebraicGeometry:AProblemSolvingApproach(2013,co-authored)and
ElectricityandMagnetismforMathematicians(2015).

All the Math You Missed
But Need to Know for Graduate School
second edition
Thomas A. Garrity
WilliamsCollege,Williamstown,MA
FiguresbyLoriPedersen
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316518403
DOI:10.1017/9781108992879
©ThomasA.Garrity2001,2021
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
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Firstpublished2001
SecondEdition2021
PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall
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DedicatedtotheMemoryof
RobertMizner

Contents
Preface pagexiii
OntheStructureofMathematics xix
BriefSummariesofTopics xxiii
0.1 LinearAlgebra xxiii
0.2 RealAnalysis xxiii
0.3 DifferentiatingVector-ValuedFunctions xxiii
0.4 PointSetTopology xxiii
0.5 ClassicalStokes’Theorems xxiv
0.6 DifferentialFormsandStokes’Theorem xxiv
0.7 CurvatureforCurvesandSurfaces xxiv
0.8 Geometry xxiv
0.9 CountabilityandtheAxiomofChoice xxv
0.10 ElementaryNumberTheory xxv
0.11 Algebra xxv
0.12 AlgebraicNumberTheory xxv
0.13 ComplexAnalysis xxv
0.14 AnalyticNumberTheory xxvi
0.15 LebesgueIntegration xxvii
0.16 FourierAnalysis xxvii
0.17 DifferentialEquations xxvii
0.18 CombinatoricsandProbabilityTheory xxvii
0.19 Algorithms xxviii
0.20 CategoryTheory xxviii
1 LinearAlgebra 1
1.1 Introduction 1
1.2 TheBasicVectorSpaceRn 1
1.3 VectorSpacesandLinearTransformations 4
1.4 Bases,Dimension,andLinearTransformationsasMatrices 6
1.5 TheDeterminant 9
1.6 TheKeyTheoremofLinearAlgebra 12
1.7 SimilarMatrices 13
1.8 EigenvaluesandEigenvectors 15
1.9 DualVectorSpaces 19
1.10 Books 20
Exercises 21
viii Contents
2 (cid:2) andδ RealAnalysis 23
2.1 Limits 23
2.2 Continuity 25
2.3 Differentiation 26
2.4 Integration 28
2.5 TheFundamentalTheoremofCalculus 31
2.6 PointwiseConvergenceofFunctions 35
2.7 UniformConvergence 36
2.8 TheWeierstrassM-Test 39
2.9 Weierstrass’Example 40
2.10 Books 43
Exercises 44
3 CalculusforVector-ValuedFunctions 46
3.1 Vector-ValuedFunctions 46
3.2 LimitsandContinuityofVector-ValuedFunctions 47
3.3 DifferentiationandJacobians 48
3.4 TheInverseFunctionTheorem 52
3.5 TheImplicitFunctionTheorem 54
3.6 Books 58
Exercises 59
4 PointSetTopology 61
4.1 BasicDefinitions 61
4.2 TheStandardTopologyonRn 63
4.3 MetricSpaces 70
4.4 BasesforTopologies 71
4.5 ZariskiTopologyofCommutativeRings 72
4.6 Books 75
Exercises 76
5 ClassicalStokes’Theorems 78
5.1 PreliminariesaboutVectorCalculus 79
5.1.1 VectorFields 79
5.1.2 ManifoldsandBoundaries 80
5.1.3 PathIntegrals 84
5.1.4 SurfaceIntegrals 88
5.1.5 TheGradient 90
5.1.6 TheDivergence 90
5.1.7 TheCurl 91
5.1.8 Orientability 92
5.2 TheDivergenceTheoremandStokes’Theorem 92
5.3 APhysicalInterpretationoftheDivergenceTheorem 94
5.4 APhysicalInterpretationofStokes’Theorem 96
Contents ix
5.5 SketchofaProofoftheDivergenceTheorem 97
5.6 SketchofaProofofStokes’Theorem 102
5.7 Books 105
Exercises 105
6 DifferentialFormsandStokes’Theorem 107
6.1 VolumesofParallelepipeds 107
6.2 DifferentialFormsandtheExteriorDerivative 111
6.2.1 Elementaryk-Forms 111
6.2.2 TheVectorSpaceofk-Forms 114
6.2.3 RulesforManipulatingk-Forms 115
6.2.4 Differentialk-FormsandtheExteriorDerivative 118
6.3 DifferentialFormsandVectorFields 120
6.4 Manifolds 122
6.5 TangentSpacesandOrientations 128
6.5.1 TangentSpacesforImplicitandParametricManifolds 128
6.5.2 TangentSpacesforAbstractManifolds 129
6.5.3 OrientationofaVectorSpace 130
6.5.4 OrientationofaManifoldandItsBoundary 131
6.6 IntegrationonManifolds 133
6.7 Stokes’Theorem 135
6.8 Books 138
Exercises 138
7 CurvatureforCurvesandSurfaces 141
7.1 PlaneCurves 141
7.2 SpaceCurves 144
7.3 Surfaces 148
7.4 TheGauss–BonnetTheorem 153
7.5 Books 154
Exercises 154
8 Geometry 156
8.1 EuclideanGeometry 156
8.2 HyperbolicGeometry 158
8.3 EllipticGeometry 161
8.4 Curvature 162
8.5 Books 163
Exercises 163
9 CountabilityandtheAxiomofChoice 165
9.1 Countability 165
9.2 NaiveSetTheoryandParadoxes 169
9.3 TheAxiomofChoice 171
9.4 Non-measurableSets 172