Download ebooks file Essentials of mathematical methods in science and engineering 2nd edition s. se

Essentials of Mathematical Methods in Science and Engineering 2nd Edition S. Selcuk Bayin

Visit to download the full and correct content document: https://ebookmass.com/product/essentials-of-mathematical-methods-in-science-and-e ngineering-2nd-edition-s-selcuk-bayin/

More products digital (pdf, epub, mobi) instant download maybe you interests ...

Mathematical Methods and Models in Composites, 2nd Edition Manti´`

https://ebookmass.com/product/mathematical-methods-and-models-incomposites-2nd-edition-manti/

Simulation and optimization in process engineering : the benefit of mathematical methods in applications of the chemical industry 1st Edition Michael Bortz

https://ebookmass.com/product/simulation-and-optimization-inprocess-engineering-the-benefit-of-mathematical-methods-inapplications-of-the-chemical-industry-1st-edition-michael-bortz/

Bioseparations Science and Engineering (Topics in Chemical Engineering) 2nd Edition – Ebook PDF Version

https://ebookmass.com/product/bioseparations-science-andengineering-topics-in-chemical-engineering-2nd-edition-ebook-pdfversion/

Essentials of Anatomy & Physiology, 2nd Edition Kenneth S. Saladin

https://ebookmass.com/product/essentials-of-anatomyphysiology-2nd-edition-kenneth-s-saladin/

Foundations of Geophysical Electromagnetic Theory and Methods 2nd Edition Michael S. Zhdanov

https://ebookmass.com/product/foundations-of-geophysicalelectromagnetic-theory-and-methods-2nd-edition-michael-s-zhdanov/

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science: Solution Methods and Chemical Engineering Applications Bertram K.C. Chan

https://ebookmass.com/product/simultaneous-mass-transfer-andchemical-reactions-in-engineering-science-solution-methods-andchemical-engineering-applications-bertram-k-c-chan/

Schaum's Outline of Mathematical Methods for Business, Economics and Finance 2nd Edition Edition Luis Moises Pena-Levano

https://ebookmass.com/product/schaums-outline-of-mathematicalmethods-for-business-economics-and-finance-2nd-edition-editionluis-moises-pena-levano/

Mathematical Methods of Analytical Mechanics 1st Edition Henri Gouin

https://ebookmass.com/product/mathematical-methods-of-analyticalmechanics-1st-edition-henri-gouin/

Advanced Methods and Mathematical Modeling of Biofilm Mojtaba Aghajani Delavar

https://ebookmass.com/product/advanced-methods-and-mathematicalmodeling-of-biofilm-mojtaba-aghajani-delavar/

EssentialsofMathematical MethodsinScience andEngineering

EssentialsofMathematical MethodsinScience andEngineering

SECONDEDITION

Sel¸cukS¸.Bayın

InstituteofAppliedMathematics

MiddleEastTechnicalUniversity

Ankara,Turkey

Thiseditionfirstpublished2020

c 2020JohnWiley&Sons,Inc.

EditionHistory

JohnWiley&Sons(1e,2008)

Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise, exceptaspermittedbylaw.Adviceonhowtoobtainpermissiontoreusematerialfromthistitleis availableat http://www.wiley.com/go/permissions

TherightofSel¸cukS¸.Bayıntobeidentifiedastheauthorofthisworkhasbeenassertedinaccordance withlaw.

RegisteredOffice

JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,USA

EditorialOffice

111RiverStreet,Hoboken,NJ07030,USA

Fordetailsofourglobaleditorialoffices,customerservices,andmoreinformationaboutWileyproducts visitusat www.wiley.com

Wileyalsopublishesitsbooksinavarietyofelectronicformatsandbyprint-on-demand.Somecontent thatappearsinstandardprintversionsofthisbookmaynotbeavailableinotherformats.

LimitofLiability/DisclaimerofWarranty

Whilethepublisherandauthorshaveusedtheirbesteffortsinpreparingthiswork,theymakeno representationsorwarrantieswithrespecttotheaccuracyorcompletenessofthecontentsofthiswork andspecificallydisclaimallwarranties,includingwithoutlimitationanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representatives,writtensalesmaterialsorpromotionalstatementsforthiswork.Thefactthatan organization,website,orproductisreferredtointhisworkasacitationand/orpotentialsourceof furtherinformationdoesnotmeanthatthepublisherandauthorsendorsetheinformationorservices theorganization,website,orproductmayprovideorrecommendationsitmaymake.Thisworkissold withtheunderstandingthatthepublisherisnotengagedinrenderingprofessionalservices.Theadvice andstrategiescontainedhereinmaynotbesuitableforyoursituation.Youshouldconsultwitha specialistwhereappropriate.Further,readersshouldbeawarethatwebsiteslistedinthisworkmay havechangedordisappearedbetweenwhenthisworkwaswrittenandwhenitisread.Neitherthe publishernorauthorsshallbeliableforanylossofprofitoranyothercommercialdamages,including butnotlimitedtospecial,incidental,consequential,orotherdamages.

LibraryofCongressCataloging-in-PublicationData

Names:Bayın,S¸.Sel¸cuk,1951-author.

Title:Essentialsofmathematicalmethodsinscienceandengineering/S¸. Sel¸cukBayın.

Description:Secondedition. | Hoboken,NJ:Wiley,2020. | Includes bibliographicalreferencesandindex.

Identifiers:LCCN2019027661(print) | LCCN2019027662(ebook) | ISBN 9781119580249(hardback) | ISBN9781119580232(adobepdf) | ISBN 9781119580287(epub)

Subjects:LCSH:Science–Mathematics. | Science–Methodology. | Engineering mathematics.

Classification:LCCQ158.5.B392020(print) | LCCQ158.5(ebook) | DDC 501/.51–dc23

LCrecordavailableathttps://lccn.loc.gov/2019027661

LCebookrecordavailableathttps://lccn.loc.gov/2019027662

CoverDesign:Wiley

CoverImage:CourtesyofSel¸cukS¸.Bayın

Setin10/12ptCMR10bySPiGlobal,Chennai,India

PrintedintheUnitedStatesofAmerica

Tomyfather, ¨ OmerBayın

Prefacexxiii Acknowledgmentsxxix 1FunctionalAnalysis1

1.1ConceptofFunction1

1.2ContinuityandLimits3

1.3PartialDifferentiation6

1.4TotalDifferential8

1.5TaylorSeries9

1.6MaximaandMinimaofFunctions13

1.7ExtremaofFunctionswithConditions17

1.8DerivativesandDifferentialsofCompositeFunctions21

1.9ImplicitFunctionTheorem23

1.10InverseFunctions28

1.11IntegralCalculusandtheDefiniteIntegral30

1.12RiemannIntegral32

1.13ImproperIntegrals35

1.14CauchyPrincipalValueIntegrals38

1.15IntegralsInvolvingaParameter40

1.16LimitsofIntegrationDependingonaParameter44

1.17DoubleIntegrals45

1.18PropertiesofDoubleIntegrals47

1.19TripleandMultipleIntegrals48 References49 Problems49

2VectorAnalysis55

2.1VectorAlgebra:GeometricMethod55

2.1.1MultiplicationofVectors57

2.2VectorAlgebra:CoordinateRepresentation60

2.3LinesandPlanes65

2.4VectorDifferentialCalculus67

2.4.1ScalarFieldsandVectorFields67

2.4.2VectorDifferentiation69

2.5GradientOperator70

2.5.1MeaningoftheGradient71

2.5.2DirectionalDerivative72

2.6DivergenceandCurlOperators73

2.6.1MeaningofDivergenceandtheDivergence Theorem75

2.7VectorIntegralCalculusinTwoDimensions79

2.7.1ArcLengthandLineIntegrals79

2.7.2SurfaceAreaandSurfaceIntegrals83

2.7.3AnAlternateWaytoWriteLineIntegrals84

2.7.4Green’sTheorem86

2.7.5InterpretationsofGreen’sTheorem88

2.7.6ExtensiontoMultiplyConnectedDomains89

2.8CurlOperatorandStokes’sTheorem92

2.8.1OnthePlane92

2.8.2InSpace96

2.8.3GeometricInterpretationofCurl99

2.9MixedOperationswiththeDelOperator99

2.10PotentialTheory102

2.10.1GravitationalFieldofaStar105

2.10.2WorkDonebyGravitationalForce106

2.10.3PathIndependenceandExactDifferentials108

2.10.4GravityandConservativeForces109

2.10.5GravitationalPotential111

2.10.6GravitationalPotentialEnergyofaSystem113

2.10.7HelmholtzTheorem115

2.10.8ApplicationsoftheHelmholtzTheorem116

2.10.9ExamplesfromPhysics120 References123 Problems123

3GeneralizedCoordinatesandTensors133

3.1TransformationsbetweenCartesianCoordinates134

3.1.1BasisVectorsandDirectionCosines134

3.1.2TransformationMatrixandOrthogonality136

3.1.3InverseTransformationMatrix137

3.2CartesianTensors139

3.2.1AlgebraicPropertiesofTensors141

3.2.2KroneckerDeltaandthePermutationSymbol145

3.3GeneralizedCoordinates148

3.3.1CoordinateCurvesandSurfaces148

3.3.2WhyUpperandLowerIndices152

3.4GeneralTensors153

3.4.1EinsteinSummationConvention156

3.4.2LineElement157

3.4.3MetricTensor157

3.4.4HowtoRaiseandLowerIndices158

3.4.5MetricTensorandtheBasisVectors160

3.4.6DisplacementVector161

3.4.7LineIntegrals162

3.4.8AreaElementinGeneralizedCoordinates164

3.4.9AreaofaSurface165

3.4.10VolumeElementinGeneralizedCoordinates169

3.4.11InvarianceandCovariance171

3.5DifferentialOperatorsinGeneralizedCoordinates171

3.5.1Gradient171

3.5.2Divergence172

3.5.3Curl174

3.5.4Laplacian178

3.6OrthogonalGeneralizedCoordinates178

3.6.1CylindricalCoordinates179

3.6.2SphericalCoordinates184 References189 Problems189

4DeterminantsandMatrices197

4.1BasicDefinitions197

4.2OperationswithMatrices198

4.3SubmatrixandPartitionedMatrices204

4.4SystemsofLinearEquations207

4.5Gauss’sMethodofElimination208

4.6Determinants211

4.7PropertiesofDeterminants214

4.8Cramer’sRule216

4.9InverseofaMatrix221

4.10HomogeneousLinearEquations224 References225 Problems225 5LinearAlgebra233

5.1FieldsandVectorSpaces233

5.2LinearCombinations,Generators,andBases236

5.3Components238

5.4LinearTransformations241

5.5MatrixRepresentationofTransformations242

5.6AlgebraofTransformations244

5.7ChangeofBasis246

5.8InvariantsunderSimilarityTransformations247

5.9EigenvaluesandEigenvectors248

5.10MomentofInertiaTensor257

5.11InnerProductSpaces262

5.12TheInnerProduct262

5.13OrthogonalityandCompleteness265

5.14Gram–SchmidtOrthogonalization267

5.15EigenvalueProblemforRealSymmetricMatrices268

5.16PresenceofDegenerateEigenvalues270

5.17QuadraticForms276

5.18HermitianMatrices279

5.19MatrixRepresentationofHermitianOperators283

5.20FunctionsofMatrices284

5.21FunctionSpaceandHilbertSpace286

5.22Dirac’sBraandKetVectors287 References288 Problems289

6PracticalLinearAlgebra293

6.1SystemsofLinearEquations294

6.1.1MatricesandElementaryRowOperations295

6.1.2Gauss-JordanMethod295

6.1.3InformationFromtheRow-EchelonForm300

6.1.4ElementaryMatrices301

6.1.5InversebyGauss-JordanRow-Reduction302

6.1.6RowSpace,ColumnSpace,andNullSpace303

6.1.7BasesforRow,Column,andNullSpaces307

6.1.8VectorSpacesSpannedbyaSetofVectors310

6.1.9RankandNullity312

6.1.10LinearTransformations315

6.2NumericalMethodsofLinearAlgebra317

6.2.1Gauss-JordanRow-ReductionandPartial Pivoting317

6.2.2LU-Factorization321

6.2.3SolutionsofLinearSystemsbyIteration325

6.2.4Interpolation328

6.2.5PowerMethodforEigenvalues331

6.2.6SolutionofEquations333

6.2.7NumericalIntegration343 References349 Problems350

7ApplicationsofLinearAlgebra355

7.1ChemistryandChemicalEngineering355

7.1.1IndependentReactionsandStoichiometricMatrix356

7.1.2IndependentReactionsfromaSetofSpecies359

7.2LinearProgramming362

7.2.1TheGeometricMethod363

7.2.2TheSimplexMethod367

7.3LeontiefInput–OutputModelofEconomy375

7.3.1LeontiefClosedModel375

7.3.2LeontiefOpenModel378

7.4ApplicationstoGeometry381

7.4.1OrbitCalculations382

7.5EliminationTheory383

7.5.1QuadraticEquationsandtheResultant384

7.6CodingTheory388

7.6.1FieldsandVectorSpaces388

7.6.2Hamming(7,4)Code390

7.6.3HammingAlgorithmforErrorCorrection393

7.7Cryptography396

7.7.1Single-KeyCryptography396

7.8GraphTheory399

7.8.1BasicDefinition399

7.8.2Terminology400

7.8.3Walks,Trails,PathsandCircuits402

7.8.4TreesandFundamentalCircuits404

7.8.5GraphOperations404

7.8.6CutSetsandFundamentalCutSets405

7.8.7VectorSpaceAssociatedwithaGraph407

7.8.8RankandNullity409

7.8.9Subspacesin WG 410

7.8.10DotProductandOrthogonalvectors411

7.8.11MatrixRepresentationofGraphs413

7.8.12DominanceDirectedGraphs417

7.8.13GrayCodesinCodingTheory419 References419 Problems420

8SequencesandSeries425

8.1Sequences426

8.2InfiniteSeries430

8.3AbsoluteandConditionalConvergence431

8.3.1ComparisonTest431

8.3.2LimitComparisonTest431

8.3.3IntegralTest431

8.3.4RatioTest432

8.3.5RootTest432

8.4OperationswithSeries436

8.5SequencesandSeriesofFunctions438

8.6 M -TestforUniformConvergence441

8.7PropertiesofUniformlyConvergentSeries441

8.8PowerSeries443

8.9TaylorSeriesandMaclaurinSeries446

8.10IndeterminateFormsandSeries447 References448 Problems448

9ComplexNumbersandFunctions453

9.1TheAlgebraofComplexNumbers454

9.2RootsofaComplexNumber458

9.3InfinityandtheExtendedComplexPlane460

9.4ComplexFunctions463

9.5LimitsandContinuity465

9.6DifferentiationintheComplexPlane467

9.7AnalyticFunctions470

9.8HarmonicFunctions471

9.9BasicDifferentiationFormulas474

9.10ElementaryFunctions475

9.10.1Polynomials475

9.10.2ExponentialFunction476

9.10.3TrigonometricFunctions477

9.10.4HyperbolicFunctions478

9.10.5LogarithmicFunction479

9.10.6PowersofComplexNumbers481

9.10.7InverseTrigonometricFunctions483 References483 Problems484 10ComplexAnalysis491

10.1ContourIntegrals492

10.2TypesofContours494

10.3TheCauchy–GoursatTheorem497

10.4IndefiniteIntegrals500

10.5SimplyandMultiplyConnectedDomains502

10.6TheCauchyIntegralFormula503

10.7DerivativesofAnalyticFunctions505

10.8ComplexPowerSeries506

10.8.1TaylorSerieswiththeRemainder506

10.8.2LaurentSerieswiththeRemainder510

10.9ConvergenceofPowerSeries514

10.10ClassificationofSingularPoints514

10.11ResidueTheorem517 References522 Problems522

11OrdinaryDifferentialEquations527

11.1BasicDefinitionsforOrdinaryDifferentialEquations528

11.2First-OrderDifferentialEquations530

11.2.1UniquenessofSolution530

11.2.2MethodsofSolution532

11.2.3DependentVariableIsMissing532

11.2.4IndependentVariableIsMissing532

11.2.5TheCaseofSeparable f (x,y )532

11.2.6Homogeneous f (x,y )ofZerothDegree533

11.2.7SolutionWhen f (x,y )IsaRationalFunction533

11.2.8LinearEquationsofFirst-order535

11.2.9ExactEquations537

11.2.10IntegratingFactors539

11.2.11BernoulliEquation542

11.2.12RiccatiEquation543

11.2.13EquationsthatCannotBeSolvedfor y 546

11.3Second-OrderDifferentialEquations548

11.3.1TheGeneralCase549

11.3.2LinearHomogeneousEquationswithConstant Coefficients551

11.3.3OperatorApproach556

11.3.4LinearHomogeneousEquationswithVariable Coefficients557

11.3.5Cauchy–EulerEquation560

11.3.6ExactEquationsandIntegratingFactors561

11.3.7LinearNonhomogeneousEquations564

11.3.8VariationofParameters564

11.3.9MethodofUndeterminedCoefficients566

11.4LinearDifferentialEquationsofHigherOrder569

11.4.1WithConstantCoefficients569

11.4.2WithVariableCoefficients570

11.4.3NonhomogeneousEquations570

11.5InitialValueProblemandUniquenessoftheSolution571

11.6SeriesSolutions:FrobeniusMethod571

11.6.1FrobeniusMethodandFirst-orderEquations581 References582 Problems582

12Second-OrderDifferentialEquationsandSpecialFunctions589

12.1LegendreEquation590

12.1.1SeriesSolution590

12.1.2EffectofBoundaryConditions593

12.1.3LegendrePolynomials594

12.1.4RodriguezFormula596

12.1.5GeneratingFunction597

12.1.6SpecialValues599

12.1.7RecursionRelations600

12.1.8Orthogonality601

12.1.9LegendreSeries603

12.2HermiteEquation606

12.2.1SeriesSolution606

12.2.2HermitePolynomials610

12.2.3ContourIntegralRepresentation611

12.2.4RodriguezFormula612

12.2.5GeneratingFunction613

12.2.6SpecialValues614

12.2.7RecursionRelations614

12.2.8Orthogonality616

12.2.9SeriesExpansionsinHermitePolynomials618

12.3LaguerreEquation619

12.3.1SeriesSolution620

12.3.2LaguerrePolynomials621

12.3.3ContourIntegralRepresentation622

12.3.4RodriguezFormula623

12.3.5GeneratingFunction623

12.3.6SpecialValuesandRecursionRelations624

12.3.7Orthogonality624

12.3.8SeriesExpansionsinLaguerrePolynomials625 References626 Problems626

13Bessel’sEquationandBesselFunctions629

13.1Bessel’sEquationandItsSeriesSolution630

13.1.1BesselFunctions J±m (x),Nm (x), and H (1,2) m (x)634

13.1.2RecursionRelations639

13.1.3GeneratingFunction639

13.1.4IntegralDefinitions641

13.1.5LinearIndependenceofBesselFunctions642

13.1.6ModifiedBesselFunctions Im (x)and Km (x)644

13.1.7SphericalBesselFunctions jl (x),nl (x), and h(1,2) l (x)645

13.2OrthogonalityandtheRootsofBesselFunctions648 13.2.1ExpansionTheorem652

13.2.2BoundaryConditionsfortheBesselFunctions652 References656 Problems656

14PartialDifferentialEquationsandSeparationofVariables661

14.1SeparationofVariablesinCartesianCoordinates662 14.1.1WaveEquation665

14.1.2LaplaceEquation666

14.1.3DiffusionandHeatFlowEquations671

14.2SeparationofVariablesinSphericalCoordinates673

14.2.1LaplaceEquation677

14.2.2BoundaryConditionsforaSphericalBoundary678

14.2.3HelmholtzEquation682

14.2.4WaveEquation683

14.2.5DiffusionandHeatFlowEquations684

14.2.6Time-IndependentSchr¨odingerEquation685

14.2.7Time-DependentSchr¨odingerEquation685

14.3SeparationofVariablesinCylindricalCoordinates686

14.3.1LaplaceEquation688

14.3.2HelmholtzEquation689

14.3.3WaveEquation690

14.3.4DiffusionandHeatFlowEquations691 References701 Problems701 15FourierSeries705

15.1OrthogonalSystemsofFunctions705

15.2FourierSeries711

15.3ExponentialFormoftheFourierSeries712

15.4ConvergenceofFourierSeries713

15.5SufficientConditionsforConvergence715

15.6TheFundamentalTheorem716

15.7UniquenessofFourierSeries717

15.8ExamplesofFourierSeries717

15.8.1SquareWave717

15.8.2TriangularWave719

15.8.3PeriodicExtension720

15.9FourierSineandCosineSeries721

15.10ChangeofInterval722

15.11IntegrationandDifferentiationofFourierSeries723 References724

Problems724

16FourierandLaplaceTransforms727

16.1TypesofSignals727

16.2SpectralAnalysisandFourierTransforms730

16.3CorrelationwithCosinesandSines731

16.4CorrelationFunctionsandFourierTransforms735

16.5InverseFourierTransform736

16.6FrequencySpectrums736

16.7Dirac-DeltaFunction738

16.8ACasewithTwoCosines739

16.9GeneralFourierTransformsandTheirProperties740

16.10BasicDefinitionofLaplaceTransform743

16.11DifferentialEquationsandLaplaceTransforms746

16.12TransferFunctionsandSignalProcessors748

16.13ConnectionofSignalProcessors750 References753

Problems753

17CalculusofVariations757

17.1ASimpleCase758

17.2VariationalAnalysis759

17.2.1CaseI:TheDesiredFunctionisPrescribedatthe EndPoints761

17.2.2CaseII:NaturalBoundaryConditions762

17.3AlternateFormofEulerEquation763

17.4VariationalNotation765

17.5AMoreGeneralCase767

17.6Hamilton’sPrinciple772

17.7Lagrange’sEquationsofMotion773

17.8DefinitionofLagrangian777

17.9PresenceofConstraintsinDynamicalSystems779

17.10ConservationLaws783

References784

Problems784

18ProbabilityTheoryandDistributions789

18.1IntroductiontoProbabilityTheory790

18.1.1FundamentalConcepts790

18.1.2BasicAxiomsofProbability791

18.1.3BasicTheoremsofProbability791

18.1.4StatisticalDefinitionofProbability794

18.1.5ConditionalProbabilityandMultiplication Theorem795

18.1.6Bayes’Theorem796

18.1.7GeometricProbabilityandBuffon’sNeedle Problem798

18.2PermutationsandCombinations800

18.2.1TheCaseofDistinguishableBallswith Replacement800

18.2.2TheCaseofDistinguishableBallsWithout Replacement801

18.2.3TheCaseofIndistinguishableBalls802

18.2.4BinomialandMultinomialCoefficients803

18.3ApplicationstoStatisticalMechanics804

18.3.1BoltzmannDistributionforSolids805

18.3.2BoltzmannDistributionforGases807

18.3.3Bose–EinsteinDistributionforPerfectGases808

18.3.4Fermi–DiracDistribution810

18.4StatisticalMechanicsandThermodynamics811

18.4.1ProbabilityandEntropy811

18.4.2Derivationof β 812

18.5RandomVariablesandDistributions814

18.6DistributionFunctionsandProbability817

18.7ExamplesofContinuousDistributions819

18.7.1UniformDistribution819

18.7.2GaussianorNormalDistribution820

18.7.3GammaDistribution821

18.8DiscreteProbabilityDistributions821

18.8.1UniformDistribution822

18.8.2BinomialDistribution822

18.8.3PoissonDistribution824

18.9FundamentalTheoremofAverages825

18.10MomentsofDistributionFunctions826

18.10.1MomentsoftheGaussianDistribution827

18.10.2MomentsoftheBinomialDistribution827

18.10.3MomentsofthePoissonDistribution829

18.11Chebyshev’sTheorem831

18.12LawofLargeNumbers832 References833 Problems834

19InformationTheory841

19.1ElementsofInformationProcessingMechanisms844

19.2ClassicalInformationTheory846

19.2.1PriorUncertaintyandEntropyofInformation848

19.2.2JointandConditionalEntropiesofInformation851

19.2.3DecisionTheory854

19.2.4DecisionTheoryandGameTheory856

19.2.5Traveler’sDilemmaandNashEquilibrium862

19.2.6ClassicalBitorCbit866

19.2.7OperationsonCbits869

19.3QuantumInformationTheory871

19.3.1BasicQuantumTheory872

19.3.2Single-ParticleSystemsandQuantum Information878

19.3.3Mach–ZehnderInterferometer880

19.3.4MathematicsoftheMach–Zehnder Interferometer882

19.3.5QuantumBitorQbit886

19.3.6TheNo-CloningTheorem889

19.3.7EntanglementandBellStates890

19.3.8QuantumDenseCoding895

Preface

Afterayearoffreshmancalculus,thebasicmathematicstraininginscienceand engineeringisusuallycompletedduringthesecondandthethirdyearsofthe undergraduatecurriculum.Studentsareusuallyrequiredtotakeasequence ofthreecoursesonthesubjectsofadvancedcalculus,differentialequations, complexcalculus,andintroductorymathematicalphysics.Today,majorityof thescienceandengineeringdepartmentsarefindingitconvenienttouseasingle bookthatassuresuniformformalismandatopicalcoverageintunewiththeir needs.Theobjectiveof EssentialsofMathematicalMethodsinScienceand Engineering istoequipstudentswiththebasicmathematicalskillsrequiredby majorityofthescienceandengineeringundergraduateprograms.

Thebookgivesacoherenttreatmentoftheselectedtopicswithastylethat makestheessentialmathematicalskillseasilyaccessibletoamultidisciplinary audience.Sincethebookiswritteninmodularformat,eachchaptercovers itssubjectthoroughlyandthuscanbereadindependently.Thisalsomakes thebookveryusefulforself-studyandasreferenceorrefresherforscientists. Itisassumedthatthereaderhasbeenexposedtotwosemestersoffreshman calculusorhasacquiredanequivalentlevelofmathematicalmaturity.

Theentirebookcontainsasufficientamountofmaterialforathree-semester coursemeetingthreetofourhoursaweek.Respectingthedisparityofthe mathematicscoursesofferedthroughouttheworld,thetopicalcoverageand themodularstructureofthebookmakeitversatileenoughtobeadoptedfora xxiii

numberofdifferentmathematicscoursesandallowsinstructorstheflexibility toindividualizetheirownteachingwhilemaintainingtheintegrityandthe uniformityofthediscussionsfortheirstudents.

AbouttheSecondEdition

Themainaimofthisbookistomeetthedemandsofthemajorityofthe modernundergraduatephysicsandengineeringprograms.Italsoaimstopreparestudentsforasolidgraduateprogramandestablishesthegroundwork ofmygraduatetextbook MathematicalMethodsinScienceandEngineering, Wiley,secondedition,2018.Thesecondedition,whilemaintainingallthesuccessfulfeaturesofthefirstedition,includestwonewandextensivechapters (Chapters6and7)entitled PracticalLinearAlgebra and ApplicationsofLinear Algebra,respectively,andacomputerfilethatincludesMatlabcodes.

ThenewchaptersweredevelopedandusedasItaughtlinearalgebra (3hrs/week)andmathematicalmethodscourses(3+1h/wk)toengineering students.ThefileincludingtheMatlabcodesisselfexplanatorybutassumes familiaritywiththetextinthebook.Thesecodeswereusedforthelab sectionofthemathematicalmethodscourseItaughttostudentswithno priorMatlabexperience.Thesecodeswillbeavailableasopensourcein https://www.wiley.com,orin http://users.metu.edu.tr/bayin/

Inadditiontothese,numerouschangeshavebeenmadetoassureeasyreadingandsmoothflowofthecomplexmathematicalarguments.Derivationsare givenwithsufficientdetailsothatthereaderwillnotbedistractedbysearching forresultsinotherpartsofthebookorbyneedingtowritedownequations.We haveshowncarefullyselectedkeywordsinboldfaceandframedkeyresultsso thattheneededinformationcanbelocatedeasilyasthereaderscansthrough thepages.

Chapterreferencesaregivenattheendofeachchapterwiththeirfull titles.Additionalresourcesfortheinterestedreaderislistedatthebackwith respecttotheirsubjectmatter.Oursuggestedreferencesisbyallmeansnot meanttobecomplete.Nowadays,readerscanlocateadditionalreferences byasimpleinternetsearch.Inparticular,readerscanusethewebsites: http://en.wikipedia.organdhttp://scienceworld.wolfram.com/.Ofcourse, https://arxiv.org isanindispensabletoolforresearchersonanysubject.

Thisbookconcentratesonanalytictechniques.Computerprogramslike Mathematica® andMaple ™ arecapableofperformingsymbolicaswellas numericalcalculations.Eventhoughtheyareextremelyusefultoscientists,one cannotstressenoughtheimportanceofafullgraspofthebasicmathematical techniqueswiththeirintricaciesandinterdisciplinaryconnections.Onlythen theunderlyingunityandthebeautyoftheuniversebeginstoappear.There arebooksspecificallywrittenformathematicalmethodswiththeseprograms, someofwhichareincludedinourlistforfurtherreadingattheback.

Withtheirexclusivechaptersanduniformlevelofformalism,thisbook connectswithmygraduatetextbook MathematicalMethodsinScienceand

Engineering,Wiley,secondedition,2018,thusformingacompletesetspanning awiderangeoffundamentalmathematicaltechniquesforstudents,instructors, andresearchers.

SummaryoftheBook

Chapter1.FunctionalAnalysis: Thischapteraimstofillthegapbetween theintroductorycalculusandtheadvancedmathematicalanalysiscourses.It introducesthebasictechniquesthatareusedthroughoutmathematics.Limits, derivatives,integrals,extremumoffunctions,implicitfunctiontheorem,inverse functions,andimproperintegralsareamongthetopicsdiscussed.

Chapter2.VectorAnalysis: Sincemostoftheclassicaltheoriescan beintroducedintermsofvectors,wepresentaratherdetailedtreatmentof vectorsandtheirtechniques.Vectoralgebra,vectordifferentiation,gradient, divergenceandcurloperators,vectorintegration,Green’stheorem,integral theorems,andtheessentialelementsofthepotentialtheoryareamongthe topicscovered.

Chapter3.GeneralizedCoordinatesandTensors: Startingwiththe Cartesiancoordinates,wediscussgeneralizedcoordinatesystemsandtheir transformations.Basisvectors,transformationmatrix,lineelement,reciprocal basisvectors,covariantandcontravariantcomponents,differentialoperatorsin generalizedcoordinates,andintroductiontoCartesianandgeneraltensorsare amongtheotheressentialtopicsofmathematicalmethods.

Chapter4.DeterminantsandMatrices: Asystematictreatmentof thebasicpropertiesandmethodsofdeterminantsandmatricesthatare muchneededinscienceandengineeringapplicationsarepresentedherewith examples.

Chapter5.LinearAlgebra: Thischapterstartswithadiscussionof abstractlinearspaces,alsocalledvectorspaces,andcontinueswithsystemsof linearequations,innerproductspaces,eigenvalueproblems,quadraticforms, Hermitianmatrices,andDirac’sbraandketvectors.

Chapter6.PracticalLinearAlgebra: Inthepreviouschapter,weconcentrateontheabstractpropertiesoflinearalgebra.Inthischapter,weintroducelinearalgebrafromthepractitionerspointofview.Inthefirstpart,we startwithsystemsoflinearequationsanddiscussGauss-Jordanreduction, row-echelonforms,elementarymatrices,rowspace,columnspaceandnull space,rankandnullity,etc.Inthesecondpart,weintroducethenumerical methodsoflinearalgebra.Partialpivoting,LU-factorization,iterationmethod, interpolation,powermethodforeigenvalues,numericalintegration,etc.are amongtheinterestingtopicsdiscussed.Inouraccompanyingwebsite,wealso haveMatlabcodesthatthereaderscanexperimentwiththemethodsintroducedinthischapter.

Chapter7.ApplicationsofLinearAlgebra: Thischapterintroduces someoftheimportantapplicationsoflinearalgebrafromdifferentbranchesof

scienceandengineering.Wegiveexamplesfromchemicalengineering,linear programming,economics,geometry,eliminationtheory,codingtheory,cryptography,andgraphtheory.

Chapter8.SequencesandSeries: Thischapterstartswithsequences andseriesofnumbersandthenintroducesabsoluteconvergenceandtestsfor convergence.Wethenextendourdiscussiontoseriesoffunctionsandintroduce theconceptofuniformconvergence.PowerseriesandTaylorseriesarediscussed indetailwithapplications.

Chapter9.ComplexNumbersandFunctions: Afterthecomplexnumbersystemisintroducedandtheiralgebraisdiscussed,complexfunctions, complexdifferentiation,Cauchy–Riemannconditionsandanalyticfunctions arethemaintopicsofthischapter.

Chapter10.ComplexAnalysis: Weintroducethecomplexintegral theoremsanddiscussresidues,Taylorseries,andLaurentseriesalongwith theirconvergenceproperties.

Chapter11.OrdinaryDifferentialEquations: Westartwiththegeneralpropertiesofdifferentialequations,theirsolutions,andboundaryconditions.Themostcommonlyencountereddifferentialequationsinapplications areeitherfirst-orsecond-orderordinarydifferentialequations.Hence,wediscussthesetwocasesseparatelyindetailandintroducemethodsoffinding theiranalyticsolutions.Wealsostudylinearequationsofhigherorder.We finallyconcludewiththeFrobeniusmethodappliedtofirst-andsecond-order differentialequationswithinterestingandcarefullyselectedexamples.

Chapter12.Second-OrderDifferentialEquationsandSpecial Functions: Inthischapter,wediscussthreeofthemostfrequentlyencounteredsecond-orderdifferentialequationsofphysicsandengineering,thatis, Legendre,Hermite,andLaguerreequations.Westudytheseequationsin detailfromtheviewpointoftheFrobeniusmethod.Byusingtheboundary conditions,wethenshowhowthecorrespondingorthogonalpolynomial setsareconstructed.Wealsodiscusshowandunderwhatconditionsthese polynomialsetscanbeusedtorepresentageneralsolution.

Chapter13.Bessel’sEquationandBesselFunctions: Besselfunctions areamongthemostfrequentlyusedspecialfunctionsofmathematicalphysics. Sincetheirorthogonalityiswithrespecttotheirrootsandnotwithrespectto aparameterinthedifferentialequation,theyarediscussedhereseparatelyin detail.

Chapter14.PartialDifferentialEquationsandSeparationof Variables: Mostofthesecond-orderordinarydifferentialequationsof physicsandengineeringareobtainedfrompartialdifferentialequationsvia themethodofseparationofvariables.Weintroducethemostcommonly encounteredpartialdifferentialequationsofphysicsandengineeringandshow howthemethodofseparationofvariablesisusedinCartesian,spherical,and cylindricalcoordinates.Interestingexampleshelpthereaderconnectwiththe knowledgegainedinthepreviousthreechapters.

Chapter15.FourierSeries: Wefirstintroduceorthogonalsystemsof functionsandthenconcentrateontrigonometricFourierseries.Wediscuss theirconvergenceanduniquenesspropertiesalongwithspecificexamples.

Chapter16.FourierandLaplaceTransforms: Afterabasicintroductiontosignalanalysisandcorrelationfunctions,weintroduceFourier transformsandtheirinverses.WealsointroduceLaplacetransformsandtheir applicationstodifferentialequations.Wediscussmethodsoffindinginverse Laplacetransformsandtheirapplicationstotransferfunctionsandsignal processors.

Chapter17.CalculusofVariations: Weintroducebasicvariational analysisfordifferenttypesofboundaryconditions.ApplicationstoHamilton’s principleandtoLagrangianmechanicsisinvestigatedindetail.Thepresence ofconstraintsindynamicalsystemsalongwiththeinverseproblemisdiscussed withexamples.

Chapter18.ProbabilityTheoryandDistributions: Someofthe interestingtopicscoveredinthischapterincludethebasictheoryofprobability,permutations,andcombinations,applicationstostatisticalmechanics, andtheconnectionwiththermodynamics.WealsodiscussBayes’theorem, randomvariables,distributions,distributionfunctionsandprobability,fundamentaltheoremofaverages,moments,Chebyshev’stheorem,andthelawof largenumbers.

Chapter19.InformationTheory: Thefirstpartofthischapteris devotedtoclassicalinformationtheory,wherewediscusstopicsfromShannon’s theory,decisiontheory,gametheory,Nashequilibrium,andtraveler’sdilemma. ThedefinitionofCbitandoperationswiththemarealsointroduced.The secondpartofthischapterisonquantuminformationtheory.Afterageneral surveyofquantummechanics,wediscussMach-Zehnderinterferometer,Qbits, entanglement,andBellstates.Alongwiththeno-cloningtheorem,quantum cryptology,quantumdensecoding,andquantumteleportationareamong theotherinterestingtopicsdiscussedinthischapter.Thischapteriswritten withastylethatmakestheseinterestingtopicsaccessibletoawiderangeof audienceswithminimumpriorexposuretoquantummechanics.

CourseSuggestions

Chapters1–17consistofthecontentsofthethree,usuallysequentiallytaught, coremathematicalmethodscoursesthatmostscienceandengineeringdepartmentsrequire.Thesechaptersalsoconsistofthebasicmathematicalskills requiredbymajorityofthemodernundergraduatescienceandengineeringprograms.Chapters1–10canbetaughtduringthesecondyearasatwo-semester course.Duringthefirstorthesecondsemesterofthethirdyear,acoursecomposedofChapters11–17cancompletethesequence.Chapters11–14canalso beusedinaseparateone-semestercourseondifferentialequationsandspecial functions.AlongwiththeMatlabcodes,Chapters4–7canbeusedtodesign

aseparatecourseonlinearalgebrawithalabsection.Instructorscanalsouse thesecodesforin-classdemonstrationswithMatlab.

Thetwoextensivechaptersonprobabilitytheoryandinformationtheory (Chapters18and19)areamongthespecialchaptersofthebook.Eventhough mostofthemathematicalmethodstextbookshavechaptersonprobability,we havetreatedthesubjectwithastyleandlevelthatpreparesthereaderfor thefollowingchapteroninformationtheory.Wehavealsoincludedsectionson applicationstostatisticalmechanicsandthermodynamics.

Thechapteroninformationtheoryisunusualforthemathematicalmethods textbooksatboththeundergraduateandthegraduatelevels.Byselecting certainsections,Chapters18and19canbeincorporatedintotheadvanced undergraduatecurriculum.Intheirentirety,theyaremoresuitabletobeused inagraduatecourse.Sincewereviewthebasicquantummechanicsneeded,we requirenopriorexposuretoquantummechanics.Inthisregard,Chapter19 isalsodesignedtobeusefultobeginningresearchersfromawiderangeof disciplinesinscienceandengineering.

Examplesandexercisesarealwaysanintegralpartofanylearningprocess; hence,thetopicsareintroducedwithanamplenumberofexamples.To maintainthecontinuityofthediscussions,wehavecollectedproblemsat theendofeachchapter,wheretheyarepredominantlylistedinthesame orderthattheyarediscussedwithinthetext.Therefore,werecommend thattheentireproblemsectionsbereadquicklybeforetheirsolutionsare attempted.Forcommunicationsaboutthebook,wewillusethewebsite http://users.metu.edu.tr/bayin/

Sel¸cukS¸.Bayın METU/IAM

Ankara,Turkey April2019

Acknowledgments

Forthefirstedition,IwouldliketothankProf.J.P.KrischoftheUniversity ofMichiganforalwaysbeingtherewheneverIneededadviceandforsharingmyexcitementatallphasesoftheproject.MyspecialthanksgotoProf. J.C.LauffenburgerandAssoc.Prof.K.D.ScherkoskeatCanisiusCollege.Iam gratefultoProf.R.P.LanglandsoftheInstituteforAdvancedStudyatPrincetonforhissupportandforhiscordialandenduringcontributionstoMETU culture.IamindebtedtoProf.P.G.L.Leachforhisinsightfulcommentsandfor meticulouslyreadingtwoofthechapters.IamgratefultoWileyforagrantto preparethecamera-readycopy,andIwouldliketothankmyeditorSusanne Steitz-Fillerforsharingmyexcitement.Myworkonthetwobooks MathematicalMethodsinScienceandEngineering and EssentialsofMathematical MethodsinScienceandEngineering hasspannedanuninterruptedperiodof sixyears.WiththetimespentonmytwobooksinTurkishpublishedinthe years2000and2004,whichwerebasicallytheforerunnersofmyfirstbook, thisprojecthasdominatedmylifeforalmostadecade.Inthisregard,IcannotexpressenoughgratitudetomydarlingyoungscientistdaughterSumru andbelovedwifeAdalet,foralwaysbeingthereformeduringthislongand strenuousjourney,whichalsoinvolvedmanysacrificesfromthem.

Forthesecondedition,IamgratefultoProf.I.TosunoftheMiddleEast TechnicalUniversityforbringingtomyattentionapplicationsoflinearalgebra xxix

tochemistryandchemicalengineeringandfornumerousdiscussionsandfor kindlyreadingtherelevantsectionsofmybook.IalsothankProf.A.S.Umar oftheVanderbiltUniversityforcommentsandforusingmybookinhiscourse. IthankProfJ.KrischoftheUniversityofMichiganforcontinuedsupport andencouragement.IthankourChairmanProf.O.UgurattheInstituteof AppliedMathematicsatMETUforsupport.IalsothankmyeditorsMindy Okura-MarszyckiandKathleenSantolociandthepublicationteamatWiley whoweremostcongenialandpleasanttoworkwith.Lastbutnottheleast,I amgratefultomytreasuredwifeAdaletandpreciousdaughterSumru,whois nowafullfledgedscientistinthefieldofstemcellresearch,fortheireverlasting loveandcare.

CHAPTER1

FUNCTIONALANALYSIS

Functionalanalysisisthebranchofmathematicsthatdealswithspacesof functionsandthetransformationpropertiesoffunctionsbetweenfunction spacesintermsofoperators.Sincetheseoperatorscouldbedifferential orintegral,itmakesfunctionalanalysisextremelyusefulinthestudyof differentialandintegralequations.Sincethefunctionandspaceconcepts couldbeusedtorepresentmanydifferentthings,functionalanalysishas foundawiderangeofapplicationsinscienceandengineering.Itisalsoatthe veryfoundationofnumericalsimulation.Themostrudimentaryconceptof functionalanalysisisthedefinitionofafunction,whichisbasicallyaruleor amappingthatrelatesthemembersofonesetofobjectstothemembersof anotherset.Inthischapter,wediscussthebasicpropertiesoffunctionslike continuity,limit,convergence,inverse,differentiation,integration,etc.

1.1CONCEPTOFFUNCTION

Westartwithaquickreviewofthebasicconceptsof settheory.Let S be asetofobjectsofanykind:points,numbers,functions,vectors,etc.When s 1 EssentialsofMathematicalMethodsinScienceandEngineering,SecondEdition.Sel¸cukS¸.Bayın. c 2020JohnWiley&Sons,Inc.Published2020byJohnWiley&Sons,Inc.

isanelementoftheset S ,weshowitas s ∈ S .Forfinitesets,wemaydefine S bylistingitselementsas S = {s1 ,s2 ,...,sn }.Forinfinitesets, S isusually definedbyaphrasedescribingtheconditiontobeamemberoftheset,for example, S = {Allpointsonthesphereofradius R}.Whenthereisnoroom forconfusion,wemayalsowriteaninfinitesetlikethesetofalloddnumbers as S = {1, 3, 5,...}.Wheneachmemberofaset A isalsoamemberofset B , wesaythat A isa subset of B andwrite A ⊂ B .Thephrase B coversor contains A isalsoused.The union oftwosets, A ∪ B ,consistsoftheelements ofboth A and B .The intersection oftwosets, A and B ,isdefinedas A ∩ B = {Allelementscommonto A and B }.Whentwosetshavenocommonelement, theirintersectioniscalledthe nullset orthe emptyset,whichisusually shownby φ.The neighborhood ofapoint(x1 ,y1 )inthe xy -planeisthesetof allpoints(x,y )insideacirclecenteredat(x1 ,y1 )withtheradius δ :(x x1 )2 + (y y1 )2 <δ 2 .An openset isdefinedasthesetofpointswithneighborhoods entirelywithintheset.Theinteriorofacircledefinedby x2 + y 2 < 1isanopen set.A boundarypoint isapointwhoseeveryneighborhoodcontainsatleast onepointinthesetandatleastonepointthatdoesnotbelongtotheset. Theboundaryof x2 + y 2 < 1isthesetofpointsonthecircumference,thatis, x2 + y 2 =1.Anopensetplusitsboundaryisa closedset.

A function f isingeneralaruleorarelationthatuniquelyassociates membersofoneset A withthemembersofanotherset B .Theconceptof functionisessentiallythesameasthatof mapping,whichingeneralissobroad thatitallowsmathematicianstoworkwiththemwithoutanyresemblanceto thesimpleclassoffunctionswithnumericalvalues.Theset A that f acts uponiscalledthe domain,andtheset B composedoftheelementsthat f canproduceiscalledthe range.For single-valued functions,thecommon notationusedis

f : x → f (x).

Here, f standsforthefunctionorthemappingthatactsuponasinglenumber x,whichisanelementofthedomain,andproduces f (x),whichisanelement oftherange.Ingeneral, f referstothefunctionitself,and f (x)referstothe valueitreturns.However,inpractice, f (x)isalsousedtorefertothefunction itself.Inthischapter,webasicallyconcernourselveswithfunctionsthattake numericalvaluesas f (x),wherethe argument x iscalledthe independent variable.Weusuallydefineanewvariable y as y = f (x),whichiscalledthe dependentvariable

Functionswithmultiplevariables,thatis, multivariate functions,canalso bedefined.Forexample,foreachpoint(x,y )insomeregionofthe xy -plane, wemayassignauniquerealnumber f (x,y )accordingtotherule f :(x,y ) → f (x,y ).Wenowsaythat f (x,y )isafunctionoftwoindependentvariables as x and y .Inapplications, f (x,y )mayrepresentphysicalpropertieslikethe temperatureorthedensitydistributionofaflatdiscwithnegligiblethickness. Definitionofafunctioncanbeextendedtocaseswithseveralindependentvariablesas f (x1 ,...,xn ),where n standsforthenumberofindependentvariables.