Applied Mathematical Methods - IITK

1 AppliedMathematicalMethods1,AppliedMathe maticalMethodsBhaskar DasguptaDepartmentof MechanicalEngineeringIndianInstituteof 13,2008 AppliedMathematicalMethods2,ContentsIPre liminaryBackgroundMatricesandLinear TransformationsOperationalFundamentalsof Linear AlgebraSystemsofLinear EquationsGaussEliminationFamilyofMethods SpecialSystemsandSpecialMethodsNumerical AspectsinLinear SystemsAppliedMathematicalMethods3,Conte ntsIIEigenvaluesandEigenvectorsDiagonali zationandSimilarity TransformationsJacobiandGivensRotationMe thodsHouseholderTransformationandTridiag onalMatricesQRDecompositionMethodEigenva lueProblemofGeneralMatricesSingular ValueDecompositionVector Spaces:FundamentalConcepts*AppliedMathem aticalMethods4,ContentsIIIT opicsinMultivariateCalculusVector Analysis:CurvesandSurfacesScalar andVector FieldsPolynomialEquationsSolutionofNonli near EquationsandSystemsOptimization:Introduc tionMultivariateOptimizationMethodsofNon linear Optimization*AppliedMathematicalMethods5 ,ContentsIVConstrainedOptimizationLinear andQuadraticProgrammingProblems*Interpol ationandApproximationBasicMethodsofNumer icalIntegrationAdvancedTopicsinNumerical Integration*NumericalSolutionofOrdinaryD i erentialEquationsODES olutions.

2 AdvancedIssuesExistenceandUniquenessTheo ryAppliedMathematicalMethods6,ContentsVF irstOrderOrdinaryDi erentialEquationsSecondOrderLinear HomogeneousODE'sSecondOrderLinear Non-HomogeneousODE'sHigherOrderLinear ODE'sLaplaceTransformsODES ystemsStability ofDynamicSystemsSeriesSolutionsandSpecia lFunctionsAppliedMathematicalMethods7,Co ntentsVISturm-LiouvilleTheoryFourierSeri esandIntegralsFourierTransformsMinimaxAp proximation*PartialDi erentialEquationsAnalyticFunctionsIntegr alsintheComplexPlaneSingularitiesofCompl exFunctionsAppliedMathematicalMethods8,C ontentsVIIV ariationalCalculus*EpilogueSelectedRefer encesAppliedMathematicalMethodsPrelimina ry Background9,Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundOutlinePreliminaryBackgroundTheme oftheCourseCourseContentsSourcesfor MoreDetailedStudyLogisticStrategyExpecte dBackgroundAppliedMathematicalMethodsPre liminary Background10.

3 Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundThemeof theCourseTo developa rmmathematicalbackgroundnecessaryfor graduatestudiesandresearchIa fast-pacedrecapitulationofUGmathematicsI extensionwithsupplementaryadvancedideasf or a matureandforwardorientationIexposureandh ighlightingofinterconnectionsTopre-emptn eedsofthefuturechallengesItrade-o betweensu cientandreasonableItargetmid-spectrummaj orityofstudentsNotablebene ciaries(attwo ends)Iwould-be researchersinanalytical/computationalare asIstudentswhoaretillnowsomewhatafraidof mathematicsAppliedMathematicalMethodsPre liminary Background11,Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundCourseContentsIAppliedlinear algebraIMultivariatecalculusandvector calculusINumericalmethodsIDi erentialequations+ +IComplexanalysisAppliedMathematicalMeth odsPreliminary Background12,Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundSourcesfor More DetailedStudyIf youhavethetime,needandinterest,thenyouma y consultIindividualbooksonindividualtopic s.

4 Ianother\umbrella"volume,like Kreyszig,McQuarrie,O'Neilor WylieandBarrett;Ia good bookofnumericalanalysisor scienti ccomputing,likeActon,Heath, Hildebrand,KrishnamurthyandSen,Pressetal , StoerandBulirsch;Ifriends, Background13,Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundLogisticStrategyIStudyinthegivens equence, \mathematicsbooks" , , a MATLAB tutorialsession?AppliedMathematicalMetho dsPreliminary Background14,Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundLogisticStrategyTutorial PlanChapterSelectionTutorialChapterSelec tionTutorial22,33261,2,4,6432,4,5,64,527 1,2,3,43,441,2,4,5,74,5282,5,6651,4,5429 1,2,5,6661,2,4,74301,2,3,4,5471,2,3,4231 1,21(d)81,2,3,4,64321,3,5,7791,2,44331,2 ,3,7,88102,3,44341,3,5,65112,4,55351,3,4 3121,33361,2,44131,213711(c)

5 142,4,5,6,74381,2,3,4,55156,77392,3,4,54 162,3,4,88401,2,4,54171,2,3,66411,3,6,88 181,2,3,6,73421,3,66191,3,4,66432,3,4320 1,2,32441,2,4,7,9,107,10211,2,5,7,87451, 2,3,4,7,94,9221,2,3,4,5,63,4461,2,5,7723 1,2,33471,2,3,5,8,9,109,10241,2,3,4,5,61 481,2,4,55251,2,3,4,55 AppliedMathematicalMethodsPreliminary Background15,Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundExpectedBackgroundImoderatebackgr oundofundergraduatemathematicsI rmunderstandingofschoolmathematicsandund ergraduatecalculusTake *AppliedMathematicalMethodsPreliminary Background16,Themeof theCourseCourseContentsSourcesfor More DetailedStudyLogisticStrategyExpectedBac kgroundPointsto noteIPutine ort, :Prerequisiteproblemsets?

6 ?AppliedMathematicalMethodsMatricesandLi near Transformations17,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyOutlineM atricesandLinear TransformationsMatricesGeometryandAlgebr aLinear TransformationsMatrixTerminologyAppliedM athematicalMethodsMatricesandLinear Transformations18,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyMatrices Question:Whatisa \matrix"?Answers:Ia rectangular array ofnumbers/elements?Ia mappingf:M N!F, whereM=f1;2;3; ;mg,N=f1;2;3; ;ngandFisthesetofrealnumbersorcomplexnum bers?Question:Whatdoesa matrixdo?Explore:Withanm nmatrixA,y1=a11x1+a12x2+ +a1nxny2=a21x1+a22x2+ + +am2x2+ +amnxn9>>>=>>>;orAx=yAppliedMathematical MethodsMatricesandLinear Transformations19,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyMatrices Considerthesede nitions:Iy=f(x)Iy=f(x) =f(x1;x2; ;xn)Iyk=fk(x) =fk(x1;x2; ;xn);k=1;2; ;mIy=f(x)Iy=AxFurtherAnswer:A matrixis thede nitionof a linear vector functionof avector :Matricesdonotde nevector functionswhosecomponentsareoftheformyk=a k0+ak1x1+ak2x2+ +aknxn:AppliedMathematicalMethodsMatrice sandLinear Transformations20,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyGeometry andAlgebraLetvectorx=[x1x2x3]Tdenotea point(x1;x2.)

7 X3) :Withm=2 andn=3,y1=a11x1+a12x2+a13x3y2=a21x1+a22x 2+a23x3 :Ploty1andy2intheOY1Y2plane. 32AR2: R Co domain Domain1xy 3 XYYXX12 OOFigure:Linear transformation:schematicillustrationWhat is matrixAdoing?AppliedMathematicalMethodsM atricesandLinear Transformations21,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyGeometry andAlgebraOperatingonpointxinR3, matrixAtransformsit ,co-domainR2withreferencetothe gureandverifythatA:R3!R2ful lstherequirementsofamapping, byde matrixgivesade nitionof alinear transformationfromonevector spaceto Transformations22,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyLinear TransformationsOperateAona transformationrepresentedbyAimpliestheto tality decidetousea di erentframeof referenceOX01X02X03forR3.

8 [And,possiblyOY01Y02forR2atthesametime.] Coordinateschange, (andpossiblyyitoy0i).Now,we needa di erentmatrix,sayA0, togetbackthecorrespondenceasy0= :HowtogetthenewmatrixA0?AppliedMathemati calMethodsMatricesandLinear Transformations23,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyMatrixTe rminologyI IMatrixproductITransposeIConjugatetransp oseISymmetricandskew-symmetricmatricesIH ermitianandskew-HermitianmatricesIDeterm inantofa squarematrixIInverseofa squarematrixIAdjointofa squarematrixI AppliedMathematicalMethodsMatricesandLin ear Transformations24,MatricesGeometryandAlg ebraLinear TransformationsMatrixTerminologyPointsto noteIAmatrixde nesa linear transformationfromonevector linear transformationdependsontheselectedbases( or framesofreference).

9 2,3 AppliedMathematicalMethodsOperationalFun damentalsof Linear Algebra25,RangeandNullSpace:RankandNulli tyBasisChangeof BasisElementary TransformationsOutlineOperationalFundame ntalsofLinear AlgebraRangeandNullSpace:RankandNullityB asisChangeofBasisElementaryTransformatio nsAppliedMathematicalMethodsOperationalF undamentalsof Linear Algebra26,RangeandNullSpace:RankandNulli tyBasisChangeof BasisElementary TransformationsRangeandNullSpace:Rankand NullityConsiderA2Rm nasa mappingA:Rn!Rm;Ax=y;x2Rn;y2 , buteveryy2 Rmneednothavea (or rangespace)as subset/subspaceofco- , be non-existent,uniqueor in subset/subspaceof Linear Algebra27,RangeandNullSpace:RankandNulli tyBasisChangeof BasisElementary TransformationsRangeandNullSpace:Rankand NullityRmRnNull ( ) AORange ( ) ADomain Co domain0 AFigure:Rangeandnullspace:schematicrepre sentationQuestion:Whatisthedimensionofa vector space?

10 Linear dependenceandindependence:Vectorsx1,x2, ,xrina vector spacearecalledlinearlyindependentifk1x1+ k2x2+ +krxr=0)k1=k2= =kr=0:Range(A)=fy:y=Ax;x2 RngNull(A)=fx:x2Rn;Ax=0gRank(A)=dimRange (A)Nullity(A)=dimNull(A)AppliedMathemati calMethodsOperationalFundamentalsof Linear Algebra28,RangeandNullSpace:RankandNulli tyBasisChangeof BasisElementary TransformationsBasisTake a setofvectorsv1,v2, ,vrina vector :Givena vectorvinthevector space,canwe describe itasv=k1v1+k2v2+ +krvr=Vk;whereV=[v1v2 vr] andk=[k1k2 kr]T? , denotedas<v1;v2; ;vr>: thesubspacedescribed/generatedby a :A basisof a vector spaceis composedof an orderedminimalsetof vectors ann-dimensionalspacewillhaveexactlynmemb ers, Linear Algebra29,RangeandNullSpace:RankandNulli tyBasisChangeof BasisElementary TransformationsBasisOrthogonalbasis:fv1; v2; ;vngwithvTjvk=08j6=k:Orthonormalbasis:vT jvk= jk= 0ifj6=k1ifj= :V 1=VTorVVT=I;anddetV=+1or 1;Naturalbasis:e1= ;e2= ; ;en= :AppliedMathematicalMethodsOperationalFu ndamentalsof Linear Algebra30,RangeandNullSpace:RankandNulli tyBasisChangeof BasisElementary TransformationsChangeof BasisSupposexrepresentsa vector (point) :If we changeovertoa newbasisfc1;c2.