高等數(shù)學(xué)下袁建華lecture 8

AdvancedMathematics(II)SchoolofScience,BUPTJianhuaYuan

Section9.32FermatDifferentiationofMultivariableCompositeFunctionsandImplicitFunctionsJacobi,Jakob3PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsTheorem

arebothSupposethatandThenthecompositedifferentiableatthecorrespondingpointisalsodifferentiableatthepointfunctionisdifferentiableatthepointwhilethefunctionanditstotaldifferentialisandthenthefunctionsuandvhavecorrespondingincrementsProofLettheincrementsofthevariablesxandybeand4PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsProof(continued)SinceuandvarebothdifferentiableatSincefisdifferentiableatthewherepointcorresponding5PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsProof(continued)Then,wecomposethefunctionsuandvintothefunctionfandwhereNow,weneedonlyverifythattheaisahigher-orderinfinitesimalThatisw.r.t.ρ.6PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsProof(continued)NoticethatSinceThusthen,isbounded.Similarly,isalsobounded.Thisimpliestheresult.isbounded.7PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsBytheformulaweknowthatandMoregenerally,ifarebothdifferentiable,thenthecompositefunctionisalsodifferentiable,where8PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsUseadiagramtoshowtheprocessingtofindthepartialdifferentialsofmultivariablecompositefunctions.TheTreeofVariablesuv9PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsSolutionFindExampleLetwhereisdifferentiable.isdifferentiablebecauseThecompositefunctionThen,wehaveandarebothdifferentiable.1210PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsTherearevariousdifferentcasesformultivariablecompositefunctions.ThentheCase1Letallbedifferentiable.Thencompositefunctionisafunctionofonevariablewhichiscalledthetotalderivative

ofthecompositefunctionzwithrespecttox.wehave11PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsCase2ThentheLetbothbedifferentiable.isalsodifferentiable,havingonecompositefunctionThen,wehaveintermediatevariablesandthreeindependentvariable.12PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsExampleProvethatLetwhereisderivable.SolutionasacompositefunctionRegardthefunctionfunctioncomposedbyandDerivationwithrespecttoxandyrespectivelygivessothat13PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsCase3ThentheLetbothbedifferentiable.isalsodifferentiable,havingonecompositefunctionThen,wehaveintermediatevariablesandtwoindependentvariable.Remarkf:三元函數(shù),自變量為x,y,u;z:復(fù)合函數(shù),u是中間變量,x,y既是中間變量又是自變量14PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsExampleLetwherethesecondorderpartialderivativesIfofthefunctionfarecontinuouswithrespecttoeachvariable.findWehaveSolutionwehaveUsingwheref1andf2expressthepartialderivativesoffunctionfwithrespecttothefirstintermediatevariableandthesecondintermediatevariablerespectively.PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsExampleThesecondorderpartialderivativesf,garecontinuouswithrespecttoeachvariable,findand15findand16PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsExampleandTransformtheexpressionsintoexpressionsinthepolarcoordinatesystem,wherehascontinuoussecondorderpartialderivatives.SolutionsothatLetand17PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsSolution(continued)asacompositefunctionThus,wecanregardthefunctionandthefunctionswhichconsistsofthefunctionandBythechainrule,weobtainandwealsohave18PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsSolution(continued)Bysubstituting,wehaveThen19PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsSolution(continued)Tofindthesecondorderderivatives,wehaveSimilarly,wealsohavetheexpressionfor20PartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsSolution(continued)Thus,weobtainthat21InvarianceofthetotaldifferentialformBytherepresentationofthetotaldifferential,weseethatSincethenTheinvarianceoffirstorderdifferentialform22InvarianceofthetotaldifferentialformRationaloperationrulesfortotaldifferentialsExample

Find23RationaloperationrulesfortotaldifferentialsExampleisdifferentiable,findthepartialderivativesofIfthefunctionSolutionBytheinvarianceofthetotaldifferentialformwehavethen24DifferentiationofImplicitFunctions(隱函數(shù))DefinedByOneEquationForanequationifthereexistsafunctionofnvariablessuchthatwhenwesubstituteitintotheequation,theequationesanidentityiscalledanimplicitfunctiondeterminedbytheequation.thenDifferentiationofImplicitFunctionsDefinedByOneEquationTheorem(Existenceofimplicitfunctions隱函數(shù)存在定理)(2)BothpartialderivativesofthefunctionFarecontinuousinasatisfiestheconditions:Supposethatthefunction(3)Then(1)neighborhoodofthepointandina(1)Thereexistsoneandonlyonefunctionhasacontinuousderivativeinthe(2)Thefunctionsuchthatneighbourhoodofthepointandneighbourhood2526DifferentiationofImplicitFunctionsDefinedByOneEquationTheproofofthistheoremisbeyondthescopeofthisbook.Wejustwithacontinuousderivative.determinesaimplicitfunctiondeducetheformulaundertheassumptionthattheequationweobtainSubstitutingintoDifferentiatebothsidesoftheaboveidentityusingthechainrule,wehavethereexistsaneighbourhoodSinceiscontinuousandthussuchthatin27DifferentiationofImplicitFunctionsDefinedByOneEquationisdeterminedbySimilarly,ifthefunctionoftwovariablesinthreevariables,thenwehaveanequationDifferentiatebothsidesoftheaboveidentity,wehavesothatSupposethatthefunctionz=z(x,y)isdeterminedbytheequationFindthetotaldifferentialdzatthepoint(1,0,-1).28DifferentiationofImplicitFunctionsDefinedByOneEquationExampleSolution(I)UsingtheformulaforderivationSolution(II)Usingtheinvarianceofthetotaldifferentialform29DifferentiationofImplicitFunctionsDefinedByOneEquationSolutionObviously,allthefirstorderLethascontinuousfirstExampleSupposethatthefunctionisdeterminedbyorderpartialderivativesandthefunctionconstants.Findwherea,bandcarealltheequationexistandpartialderivativesofthecompositefunctionwehavearecontinuous.soLet30DifferentiationofImplicitFunctionsDefinedByMoreThanOneEquationConsiderthesystemoftwoequationsoffunctionsIngeneral,twooftheThesetwoequationscontainfourvariables.Iftheycandeterminetwovariablescanbedeterminedbytheothers.functionsoftwovariableswithcontinuouspartialderivativessothatDifferentiatebothsidesofaboveidentities,wehave31DifferentiationofImplicitFunctionsDefinedByMoreThanOneEquationThesetwoequationsformalinearalgebraicsystemforthetwounknownandquantitiesIfthedeterminantofcoefficientsthen,bytheCramer’srule,weobtainBythesameway,wehaveJacobi,Jakob(1804-1851)Germanmathematician32DifferentiationofImplicitFunctionsDefinedByMoreThanOneEquationExampleFindthepartialderivativesoftheimplicitfunctionsdeterminedbytheequationsandSolution(I)Byformulasofderivativesoftheimplicitfunctions,ifweobtainSimilarly,wehaveand33DifferentiationofImplicitFunctionsDefinedByMoreThanOneEquationSolution(II)Findthetotaldifferentialstobothsidesoftheequations.wehaveWhileExampleFindthepartialderivativesoftheimplicitfunctionsdeterminedbytheequationsand34ReviewPartialDerivativesandTotalDifferentialsofMultivariableCompositeFunctionsDifferentiationofImplicitFunctionsDefinedByOneEquationDifferentiationofImplicitFunctionsDefinedByMoreThanOneEquation補(bǔ)充說明復(fù)合函數(shù)與非復(fù)合函數(shù)其中f:三元函數(shù),自變量為u,v,w;z:復(fù)合函數(shù),u,v,w是中間變量,x,y是自變量求偏導(dǎo)數(shù)時(shí)分清自變量和中間變量其中f的偏導(dǎo)數(shù)：z的偏導(dǎo)數(shù)：可以分別記為或?qū)χ虚g變量的導(dǎo)數(shù)對(duì)最終自變量的導(dǎo)數(shù)35補(bǔ)充說明f:三元函數(shù),自變量為u,x,y;z:復(fù)合函數(shù),u,是中間變量,x,y既是中間又是自變量求偏導(dǎo)數(shù)時(shí)分清自變量和中間變量其中f的偏導(dǎo)數(shù)：可以分別記為或z的偏導(dǎo)數(shù)：對(duì)中間變量的導(dǎo)數(shù)對(duì)最終自變量的導(dǎo)數(shù)不一定等于不一定等于36補(bǔ)充說明求偏導(dǎo)數(shù)時(shí)分清自變量和中間變量其中不一定等于不一定等于Exampleswhere

where37Section9.438FermatDerivativesandDifferentialsofVector-valuedFunctions39Vector-ValuedFunctionsJustaswedidforplanarcurvesbefore,totractaparticlemovinginspace,werunavectorrfromtheorigintotheparticleandstudythechangesinr.Iftheparticle’spositioncoordinatesaretwice-differentiablefunctionsoftime,thensoisr,andwecanfindtheparticle’svelocityandaccelerationvectorsatanytimebydifferentiatingr.Conversely,ifwehaveenoughinformationabouttheparticle’sinitialvelocityandposition,wecanfindrasafunctionoftimebyintegration.40SpaceCurvesWhenaparticlemovesthroughspaceduringatimeintervalI,wethinkoftheparticle’scoordinatesasfunctionsdefinedonI:makeupthecurveinspaceThepointsTheequationsandintervalinlastthatwecalltheparticle’spath.equationparametrizethecurve.AcurveinspacecanalsobewroteinThevectorvectorform.definesrasavectorfunctionoftherealvariabletontheintervalI.Moregenerally,avectorfunctionorvector-valuedfunctiononadomainsetDisarulethatassignsavectorinspacetoeachelementinD.41LimitsandContinuityDefinitionLimitandContinuity,thenIfinitsdomainifAvectorfunctioniscontinuousatapointThefunctioniscontinuousifitiscontinuousateverypointinitsdomain.ExamplethenSupposethat42DerivativesisdifferentiableifitisdifferentiableateverypointiscontinuousandThatisiff,gandhhavecontinuousfirstderivativesthatareAvectorfunctionDefinitionDerivativeataPoint

isdifferentiable

atThevectorfunctionThederivativeist