Mathematics for Physics

《Mathematics for Physics》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、MathematicsforPhysicsIMichaelStoneandPaulGoldbartPIMANDER-CASAUBONAlexandria•Florence•LondoniiCopyrightc2000-2008M.Stone,P.Goldbart.Allrightsreserved.Nopartofthismaterialcanbereproduced,storedortransmittedwithoutthewrittenpermissionoftheauthor.Forinformationcontact:MichaelStone,LoomisLabor

2、atoryofPhysics,UniversityofIllinois,1110WestGreenStreet,Urbana,IL61801,USA.PrefaceThesenoteswerepreparedforthefirstsemesterofayear-longmathematicalmethodscourseforbegininggraduatestudentsinphysics.Theemphasisisonlinearoperatorsandstressestheanalogybetweensuchoperatorsactingonfunctionspacesa

3、ndmatricesactingonfinitedimensionalspaces.Theop-eratorlanguagethenprovidesaunifiedframeworkforinvestigatingordinaryandpartialdifferentialequations,andintegralequations.Themathematicalprerequisitesforthecourseareasoundgraspofun-dergraduatecalculus(includingthevectorcalculusneededforelectricity

4、andmagnetismcourses),linearalgebra(themorethebetter),andcompetenceatcomplexarithmetic.Fouriersumsandintegrals,aswellasbasicordinarydifferentialequationtheoryreceiveaquickreview,butitwouldhelpifthereaderhadsomepriorexperiencetobuildon.Contourintegrationisnotrequired.iiiivPREFACEContentsPrefa

5、ceiii1CalculusofVariations11.1Whatisitgoodfor?.......................11.2Functionals............................21.2.1Thefunctionalderivative.................21.2.2TheEuler-Lagrangeequation..............31.2.3Someapplications.....................41.2.4Firstintegral.......................10

6、1.3LagrangianMechanics......................111.3.1Onedegreeoffreedom..................121.3.2Noether’stheorem....................151.3.3Manydegreesoffreedom.................181.3.4Continuoussystems....................191.4VariableEndPoints........................291.5LagrangeMultipliers..

7、.....................361.6MaximumorMinimum?.....................401.7FurtherExercisesandProblems.................422FunctionSpaces552.1Motivation.............................552.1.1Functionsasvectors...................562.2NormsandInnerProducts................