Calculus 1 to 4 (2004–2006) Axel Schuler¨ January 3, 2007 2 Contents 1 Real and Complex Numbers 11 Basics .......................................... 11 Notations ..................................... 11 SumsandProducts ................................ 12 MathematicalInduction. 12 BinomialCoefficients............................... 13 1.1 RealNumbers................................... 15 1.1.1 OrderedSets ............................... 15 1.1.2 Fields................................... 17 1.1.3 OrderedFields .............................. 19 1.1.4 Embedding of natural numbers into the real numbers . ....... 20 1.1.5 The completeness of Ê .......................... 21 1.1.6 TheAbsoluteValue............................ 22 1.1.7 SupremumandInfimumrevisited . 23 1.1.8 Powersofrealnumbers.. .. .. .. .. .. .. 24 1.1.9 Logarithms ................................ 26 1.2 Complexnumbers................................. 29 1.2.1 TheComplexPlaneandthePolarform . 31 1.2.2 RootsofComplexNumbers . 33 1.3 Inequalities .................................... 34 1.3.1 MonotonyofthePowerand ExponentialFunctions . ...... 34 1.3.2 TheArithmetic-Geometricmeaninequality . ...... 34 1.3.3 TheCauchy–SchwarzInequality . .. 35 1.4 AppendixA.................................... 36 2 Sequences and Series 43 2.1 ConvergentSequences .. .. .. .. .. .. .. .. 43 2.1.1 Algebraicoperationswithsequences. ..... 46 2.1.2 Somespecialsequences . .. .. .. .. .. .. 49 2.1.3 MonotonicSequences .. .. .. .. .. .. .. 50 2.1.4 Subsequences............................... 51 2.2 CauchySequences ................................ 55 2.3 Series ....................................... 57 3 4 CONTENTS 2.3.1 PropertiesofConvergentSeries . ... 57 2.3.2 OperationswithConvergentSeries. .... 59 2.3.3 SeriesofNonnegativeNumbers . 59 2.3.4 TheNumber e .............................. 61 2.3.5 TheRootandtheRatioTests. 63 2.3.6 AbsoluteConvergence . 65 2.3.7 DecimalExpansionofRealNumbers . 66 2.3.8 ComplexSequencesandSeries . 67 2.3.9 PowerSeries ............................... 68 2.3.10 Rearrangements.............................. 69 2.3.11 ProductsofSeries ............................ 72 3 Functions and Continuity 75 3.1 LimitsofaFunction............................... 76 3.1.1 One-sided Limits, Infinite Limits, and Limits at Infinity......... 77 3.2 ContinuousFunctions. .. 80 3.2.1 TheIntermediateValueTheorem. .. 81 3.2.2 Continuous Functions on Bounded and Closed Intervals—The Theorem about Maximum and Minimum 3.3 UniformContinuity............................... 83 3.4 MonotonicFunctions .............................. 85 3.5 Exponential, Trigonometric, and Hyperbolic Functions and their Inverses . 86 3.5.1 ExponentialandLogarithmFunctions . .... 86 3.5.2 TrigonometricFunctionsandtheirInverses . ....... 89 3.5.3 HyperbolicFunctionsandtheirInverses . ...... 94 3.6 AppendixB.................................... 95 3.6.1 MonotonicFunctionshaveOne-SidedLimits . ..... 95 3.6.2 Proofs for sin x and cos x inequalities .................. 96 3.6.3 Estimates for π .............................. 97 4 Differentiation 101 4.1 TheDerivativeofaFunction . 101 4.2 TheDerivativesofElementaryFunctions . ....... 107 4.2.1 DerivativesofHigherOrder . 108 4.3 LocalExtremaandtheMeanValueTheorem . 108 4.3.1 LocalExtremaandConvexity . 111 4.4 L’Hospital’sRule ................................ 112 4.5 Taylor’sTheorem ................................. 113 4.5.1 ExamplesofTaylorSeries . 115 4.6 AppendixC.................................... 117 5 Integration 119 5.1 TheRiemann–StieltjesIntegral . ...... 119 5.1.1 PropertiesoftheIntegral . 126 CONTENTS 5 5.2 IntegrationandDifferentiation . ....... 132 5.2.1 TableofAntiderivatives . 134 5.2.2 IntegrationRules ............................. 135 5.2.3 IntegrationofRationalFunctions . 138 5.2.4 PartialFractionDecomposition . 140 5.2.5 Other Classes of Elementary Integrable Functions . ......... 141 5.3 ImproperIntegrals ............................... 143 5.3.1 Integralsonunboundedintervals . 143 5.3.2 IntegralsofUnboundedFunctions . 146 5.3.3 TheGammafunction. .. .. .. .. .. .. .. 147 5.4 IntegrationofVector-ValuedFunctions . ......... 148 5.5 Inequalities .................................... 150 5.6 AppendixD.................................... 151 5.6.1 MoreontheGammaFunction . 152 6 Sequences of Functions and Basic Topology 157 6.1 DiscussionoftheMainProblem . 157 6.2 UniformConvergence. .. .. .. .. .. .. .. .. 158 6.2.1 DefinitionsandExample . 158 6.2.2 UniformConvergenceandContinuity . 162 6.2.3 UniformConvergenceandIntegration . 163 6.2.4 UniformConvergenceandDifferentiation . ...... 166 6.3 FourierSeries ................................... 168 6.3.1 AnInnerProductonthePeriodicFunctions . ..... 171 6.4 BasicTopology .................................. 177 6.4.1 Finite,Countable,andUncountableSets . ...... 177 6.4.2 MetricSpacesandNormedSpaces. 178 6.4.3 OpenandClosedSets .. .. .. .. .. .. .. 180 6.4.4 LimitsandContinuity .. .. .. .. .. .. .. 182 6.4.5 ComletenessandCompactness. 185 k 6.4.6 Continuous Functions in Ê ....................... 187 6.5 AppendixE .................................... 188 7 Calculus of Functions of Several Variables 193 7.1 PartialDerivatives.. .. .. .. .. .. .. .. .. 194 7.1.1 HigherPartialDerivatives . 197 7.1.2 TheLaplacian............................... 199 7.2 TotalDifferentiation. 199 7.2.1 BasicTheorems.............................. 202 7.3 Taylor’sFormula ................................. 206 7.3.1 DirectionalDerivatives . 206 7.3.2 Taylor’sFormula ............................. 208 7.4 ExtremaofFunctionsofSeveralVariables . ........ 211 6 CONTENTS 7.5 TheInverseMappingTheorem . 216 7.6 TheImplicitFunctionTheorem. 219 7.7 LagrangeMultiplierRule . 223 7.8 IntegralsdependingonParameters . ...... 225 7.8.1 Continuity of I(y) ............................ 225 7.8.2 DifferentiationofIntegrals . 225 7.8.3 ImproperIntegralswithParameters . 227 7.9 Appendix ..................................... 230 8 Curves and Line Integrals 231 8.1 RectifiableCurves................................ 231 k 8.1.1 Curvesin Ê ............................... 231 8.1.2 RectifiableCurves ............................ 233 8.2 LineIntegrals ................................... 236 8.2.1 PathIndependence ............................ 239 9 Integration of Functions of Several Variables 245 9.1 BasicDefinition.................................. 245 9.1.1 PropertiesoftheRiemannIntegral . 247 9.2 IntegrableFunctions ............................. 248 9.2.1 IntegrationoverMoreGeneralSets . 249 9.2.2 Fubini’sTheoremandIteratedIntegrals . ...... 250 9.3 ChangeofVariable ................................ 253 9.4 Appendix ..................................... 257 10 Surface Integrals 259 3 10.1 Surfaces in Ê ................................... 259 10.1.1 TheAreaofaSurface .. .. .. .. .. .. .. 261 10.2 ScalarSurfaceIntegrals. ..... 262 10.2.1 Other Forms for dS ........................... 262 10.2.2 PhysicalApplication . 264 10.3 SurfaceIntegrals ............................... 264 10.3.1 Orientation ................................ 264 10.4 Gauß’DivergenceTheorem. 268 10.5 Stokes’Theorem ................................. 272 10.5.1 Green’sTheorem . .. .. .. .. .. .. .. 272 10.5.2 Stokes’Theorem ............................. 274 10.5.3 Vector Potential and the Inverse Problem of Vector Analysis . 276 n 11 Differential Forms on Ê 279 n 11.1 The Exterior Algebra Λ(Ê ) ........................... 279 11.1.1 The Dual Vector Space V ∗ ........................ 279 11.1.2 The Pull-Back of k-forms ........................ 284 n 11.1.3 Orientation of Ê ............................. 285 CONTENTS 7 11.2 DifferentialForms. 285 11.2.1 Definition................................. 285 11.2.2 Differentiation .. .. .. .. .. .. .. .. 286 11.2.3 Pull-Back................................. 288 11.2.4 ClosedandExactForms . 291 11.3 Stokes’Theorem ................................. 293 11.3.1 Singular Cubes, Singular Chains, and the Boundary Operator . 293 11.3.2 Integration ................................ 295 11.3.3 Stokes’Theorem ............................. 296 11.3.4 SpecialCases .............................. 298 11.3.5 Applications ............................... 299 12 Measure Theory and Integration 305 12.1 MeasureTheory.................................. 305 12.1.1 Algebras, σ-algebras,andBorelSets. 306 12.1.2 AdditiveFunctionsandMeasures . 308 12.1.3 ExtensionofCountablyAdditiveFunctions . ....... 313 n 12.1.4 The Lebesgue Measure on Ê ...................... 314 12.2 MeasurableFunctions. 316 12.3 TheLebesgueIntegral . 318 12.3.1 SimpleFunctions. .. .. .. .. .. .. .. 318 12.3.2 PositiveMeasurableFunctions . 319 12.4 SomeTheoremsonLebesgueIntegrals. ...... 322 12.4.1 TheRolePlayedbyMeasureZeroSets . 322 12.4.2 The space Lp(X,µ) ............................ 324 12.4.3 TheMonotoneConvergenceTheorem . 325 12.4.4 TheDominatedConvergenceTheorem . 326 12.4.5 Application of Lebesgue’s Theorem to Parametric Integrals. 327 12.4.6 TheRiemannandtheLebesgueIntegrals . 329 12.4.7 Appendix:Fubini’sTheorem. 329 13 Hilbert Space 331 13.1 TheGeometryoftheHilbertSpace. ..... 331 13.1.1 UnitarySpaces .............................. 331 13.1.2 NormandInnerproduct . 334 13.1.3 TwoTheoremsofF.Riesz . 335 13.1.4 OrthogonalSetsandFourierExpansion . ..... 339 13.1.5 Appendix ................................. 343 13.2 BoundedLinearOperatorsinHilbertSpaces . ......... 344 13.2.1 BoundedLinearOperators . 344 13.2.2 TheAdjointOperator. 347 13.2.3 ClassesofBoundedLinearOperators . 349 13.2.4 OrthogonalProjections . 351 8 CONTENTS 13.2.5 SpectrumandResolvent