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1、2C2MultivariateCalculusMichaelD.AlderNovember13,20022Contents1Introduction52Optimisation72.1TheSecondDerivativeTest...................73ConstrainedOptimisation153.1LagrangianMultipliers......................154FieldsandForms234.1De?nitionsGalore.........................234.2Integrating1-forms(vector
2、?elds)overcurves..........304.3IndependenceofParametrisation................344.4ConservativeFields/ExactForms................374.5ClosedLoopsandConservatism.................405Green’sTheorem475.1Motivation.............................475.1.1Functionsastransformations...............475.1.2ChangeofVa
3、riablesinIntegration...........505.1.3SpinFields........................525.2Green’sTheorem(ClassicalVersion)..............5534CONTENTS5.3Spin?eldsandDi?erential2-forms...............585.3.1TheExteriorDerivative.................635.3.2ForthePureMathematicians...............705.3.3Returntothe(relativ
4、ely)mundane............725.4MoreonDi?erentialStretching.................735.5Green’sTheoremAgain.....................876Stokes’Theorem(ClassicalandModern)976.1Classical..............................976.2Modern..............................1036.3Divergence.............................1167FourierTheor
5、y1237.1VariousKindsofSpaces.....................1237.2FunctionSpaces.........................1287.3Applications...........................1327.4FiddlyThings..........................1357.5OddandEvenFunctions....................1427.6FourierSeries...........................1437.7Di?erentiationandInte
6、grationofFourierSeries........1507.8Functionsofseveralvariables..................1518PartialDi?erentialEquations1558.1Introduction............................1558.2TheDi?usionEquation......................1598.2.1Intuitive..........................1598.2.2SayingitinAlgebra...................1628.3La
7、place’sEquation........................165CONTENTS58.4TheWaveEquation........................1698.5Schr¨odinger’sEquation......................1738.6TheDirichletProblemforLaplace’sEquation.........174
