Multivariate_Calculus-Alder.pdf

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2C2MultivariateCalculus
MichaelD.Alder
November13,2002
2
Contents
1Introduction 5
2Optimisation 7
2.1TheSecondDerivativeTest................... 7
3ConstrainedOptimisation 15
3.1LagrangianMultipliers......................15
4FieldsandForms 23
4.1DefinitionsGalore.........................23
4.2Integrating1-forms(vectorfields)overcurves..........30
4.3IndependenceofParametrisation................34
4.4ConservativeFields/ExactForms................37
4.5ClosedLoopsandConservatism.................40
5Green’sTheorem 47
5.1Motivation.............................47
5.1.1Functionsastransformations...............47
5.1.2ChangeofVariablesinIntegration...........50
5.1.3SpinFields........................52
5.2Green’sTheorem(ClassicalVersion)..............55
3
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4 CONTENTS
5.3SpinfieldsandDierential2-forms...............58
5.3.1TheExteriorDerivative.................63
5.3.2ForthePureMathematicians...............70
5.3.3Returntothe(relatively)mundane............72
5.4MoreonDierentialStretching.................73
5.5Green’sTheoremAgain.....................87
6Stokes’Theorem(ClassicalandModern) 97
6.1Classical..............................97
6.2Modern..............................103
6.3Divergence.............................116
7FourierTheory 123
7.1VariousKindsofSpaces.....................123
7.2FunctionSpaces.........................128
7.3Applications...........................132
7.4FiddlyThings..........................135
7.5OddandEvenFunctions....................142
7.6FourierSeries...........................143
7.7DierentiationandIntegrationofFourierSeries........150
7.8Functionsofseveralvariables..................151
8PartialDierentialEquations 155
8.1Introduction............................155
8.2TheDiusionEquation......................159
8.2.1Intuitive..........................159
8.2.2SayingitinAlgebra...................162
8.3Laplace’sEquation........................165
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CONTENTS 5
8.4TheWaveEquation........................169
8.5Schr¨odinger’sEquation......................173
8.6TheDirichletProblemforLaplace’sEquation.........174
8.7LaplaceonDisks.........................181
8.8SolvingtheHeatEquation....................185
8.9SolvingtheWaveEquation....................191
8.10AndinConclusion..........................194
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