Several Variable calculus

244 Pages·2022·0.707 MB·English

by  PSC

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Several Variable CIalculus
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Partha Sarathi Chakraborty
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February 9, 2021
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Contents U
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0 Agenda 1
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0.1 ThingsTwe need to decide . . . . . . . . . . . 1
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0.2 What do we do in this course? . . . . . . . . . 3
1 Normed Linear Spaces 5
1.1 Metric Spaces . . . . . . . . . . . . . . . . . . . 6
1.2 Examples from geometry. . . . . . . . . . . . . 7
[Lecture Notes of P.S.Chakraborty]
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1.2.1 Euclidean Metric . . . . . . . . . . . . . 7
1.2.2 Spherical metric . . . . . . . . . . . . . 11
1.3 Normed Linear Spaces . . . . . . . . . . .N . . . 15
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1.4 Continuity . . . . . . . . . . . . . . T. . . . . . . 17
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1.5 Topology of a metric space . . . . . . . . . . . 18
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1.6 Compact subsets of a finSite dimensional vector
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space . . . . . . . . .D. . . . . . . . . . . . . . . 20
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1.7 Continuous Linear Maps . . . . . . . . . . . . . 22
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1.8 Bounded LFinear Maps . . . . . . . . . . . . . . 22
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1.9 EquivalenceofNormsonFiniteDimensionalVec-
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tor Spaces . . . . . . . . . . . . . . . . . . . . . 25
2 Completeness, Contractions, Consequences 27
2.1 Lipschitz Maps . . . . . . . . . . . . . . . . . . 29
2.2 The Classification Question . . . . . . . . . . . 31
[Lecture Notes of P.S.Chakraborty]
[iii]
2.3 Series in Banach Spaces . . . . . . . . . . . . . 33
2.4 Contraction Principle . . . . . . . . . . . . . . 36
2.4.1 Applications of Contraction TheorNem . 38
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3 Exponential, Logarithm and other Functions 43
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3.1 Bounded Linear maps as aTnormed linear space 44
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3.2 The General Linear Group of a Banach Space . 47
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3.3 The exponentiRal of a linear map. . . . . . . . . 51
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3.4 LogaritThm and other functions . . . . . . . . . 56
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4 Differentiation 59
4.1 The Gˆateaux Derivative . . . . . . . . . . . . . 60
4.2 The Fr´echet Derivative . . . . . . . . . . . . . . 61
4.3 Gˆateaux Differential as Velocity . . . . . . . . . 64
[Lecture Notes of P.S.Chakraborty]
[iv]
4.4 ElementaryPropertiesofGˆateaux/Fr´echetDeriva-
tive. . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4.1 Application: ConservationofAngularMo-
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mentum . . . . . . . . . . . . . . . . . . 73
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4.5 Comparison of Gˆateaux and Fr´echet derivative 74
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5 The Jacobian (Operator) MatriTx 79
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5.1 Differentiating Maps to a Direct Sum. . . . . . 80
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5.2 DifferentiatingRMapsdefinedonanOpenSubset
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of a DirecFt Sum . . . . . . . . . . . . . . . . . . 84
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5.3 CombOining the Two: Maps from an Open Sub-
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set of a Direct Sum to a Direct Sum . . . . . . 88
6 The Integral 91
6.1 The Cauchy Integral . . . . . . . . . . . . . . . 92
6.2 Characterization of Regulated Functions . . . . 95
[Lecture Notes of P.S.Chakraborty]
[v]
6.3 Characterization of the Cauchy Integral . . . . 97
6.4 The Riemann Integral . . . . . . . . . . . . . . 99
6.5 Properties of the Integral . . . . . . . . .N . . . 101
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7 Taylor’s Formula U 105
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7.1 Fundamental Theorem of Calculus . . . . . . . 105
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7.2 Symmetries of HigherDDerivatives . . . . . . . . 108
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7.3 Taylor’s Formula . . . . . . . . . . . . . . . . . 109
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8 ApplicationsTI: Linear Lie Groups 113
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8.1 Properties of the Exponential Map . . . . . . . 114
8.2 Lie Algebra of a Linear Lie Group . . . . . . . 115
8.3 Lie Algebras of Some Linear Lie
Groups. . . . . . . . . . . . . . . . . . . . . . . 121
8.4 Lie Group Lie Algebra Correspondence. . . . . 122
[Lecture Notes of P.S.Chakraborty]
[vi]
8.5 BCH Formula . . . . . . . . . . . . . . . . . . . 131
8.6 Integrating Lie Algebra Homomorphisms . . . . 139
8.7 Simply Connected Linear Lie Groups . . .N . . . 148
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9 Isomorphism Theorems U 151
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9.1 Inverse Function Theorem . . . . . . . . . . . . 152
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9.1.1 Corollaries of IDnverse Function Theorem 155
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9.2 Implicit Function Theorem . . . . . . . . . . . 157
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10 Ordinary DiTfferential Equations 161
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10.1 Reduction to time/parameter independent case 164
10.1.1 The time dependent case . . . . . . . . 164
10.1.2 The time and parameter dependent case 165
10.2 ExistenceandUniquenessofFlowsforLipschitz
Vector Fields . . . . . . . . . . . . . . . . . . . 166
[Lecture Notes of P.S.Chakraborty]
[vii]
10.3 Differentiability of Dependence on Initial Con-
ditions . . . . . . . . . . . . . . . . . . . . . . . 168
10.4 The Flow Equation . . . . . . . . . . . . . . . . 172
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10.5 Vector Fields as Derivations . . . . . . O. . . . . 174
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11 Applications: Optimization 179
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11.1 Unconstrained Otimizaton . . . . . . . . . . . . 179
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11.2 Critical Points andEGradient Flow . . . . . . . 181
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11.3 Constrained Optimization: Necessary Condition 185
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11.4 Constrained Optimization: Sufficient Condition 187
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11.5 GeNometric Description of Lagrange Rule . . . . 190
12 Change of Variables in Multiple Integrals 191
13 Multilinear Algebra 193
13.1 tensor Products . . . . . . . . . . . . . . . . . . 194
[Lecture Notes of P.S.Chakraborty]
[viii]
13.2 G-Spaces . . . . . . . . . . . . . . . . . . . . . 198
13.3 Alternating/Exterior/Wedge Product of Vector
Spaces . . . . . . . . . . . . . . . . . . . . . . . 200
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13.4 The Wedge/Antisymmetric product Map . . . 204
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13.4.1 A Second Approach to Wedge Product
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Map . . . . . . . . .R. . . . . . . . . . . 205
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13.5 Pairing Between AlterInating Forms and Anti-
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symmetric ProductE. . . . . . . . . . . . . . . . 206
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Appendices
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Appendix AN Axiom of Choice and All That! 211
Appendix B Hahn-Banach Theorems 217
Appendix C Homotopy 223
C.1 Fundamental Group . . . . . . . . . . . . . . . 228
[Lecture Notes of P.S.Chakraborty]