Mathematics for Engineering T u r y n A D V A N C E D ADVANCED ENGINEERING MATHEMATICS E N G I N E E R I N G Beginning with linear algebra and later expanding into calculus of varia- tions, Advanced Engineering Mathematics provides accessible and comprehensive mathematical preparation for advanced undergraduate and beginning graduate students taking engineering courses. This book M AT H E M AT I C S offers a review of standard mathematics coursework while effectively MEA integrating science and engineering throughout the text. It explores the use of engineering applications, carefully explains links to engineering N practice, and introduces the mathematical tools required for understand- A D ing and utilizing software packages. G • Provides comprehensive coverage of mathematics used by T engineering students V • Combines stimulating examples with formal exposition and HI provides context for the mathematics presented • Contains a wide variety of applications and homework problems N • Includes over 300 figures, more than 40 tables, and over E A 2 1500 equations z • Introduces useful Mathematica™ and MATLAB® procedures E M • Presents faculty and student ancillaries, including an online student solutions manual, full solutions manual for instructors, N E 0 and full-color figure slides for classroom presentations A Advanced Engineering Mathematics covers ordinary and partial R differential equations, matrix/linear algebra, Fourier series and transforms, C T and numerical methods. Examples include the singular value decomposi- -2 tion for matrices, least squares solutions, difference equations, the I z-transform, Rayleigh methods for matrices and boundary value problems, I E the Galerkin method, numerical stability, splines, numerical linear algebra, N curvilinear coordinates, calculus of variations, Liapunov functions, C -1 -3 controllability, and conformal mapping. GD 0 -2 S This text also serves as a good reference book for students seeking y x additional information. It incorporates Short Takes sections, describing a 1 -1 more advanced topics to readers, and Learn More about It sections with 0 direct references for readers wanting more in-depth information. L a r r y T u r y n K11552 A D V A N C E D E N G I N E E R I N G M AT H E M AT I C S L a r r y T u r y n Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130801 International Standard Book Number-13: 978-1-4822-1939-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. 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CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface.............................................................................................. xix Acknowledgments ............................................................................... xxv 1. LinearAlgebraicEquations,Matrices,andEigenvalues............................ 1 1.1 SolvingSystemsandRowEchelonForms ........................................ 1 1.1.1 Matrices ........................................................................ 2 1.1.2 AugmentedMatrices......................................................... 5 1.1.3 RowReducedEchelonForm(RREF)....................................... 6 1.1.4 Problems....................................................................... 13 1.2 MatrixAddition,Multiplication,andTranspose................................. 16 1.2.1 SpecialKindsofMatrices.................................................... 21 1.2.2 PowersofaMatrix............................................................ 23 1.2.3 Transpose...................................................................... 24 1.2.4 ElementaryMatrices ......................................................... 24 1.2.5 Problems....................................................................... 25 1.3 HomogeneousSystems,SpanningSet,andBasicSolutions.................... 26 1.3.1 Problems....................................................................... 32 1.4 SolutionsofNonhomogeneousSystems .......................................... 33 1.4.1 Problems....................................................................... 36 1.5 InverseMatrix......................................................................... 37 1.5.1 RowReductionAlgorithmforConstructingtheInverse............... 40 1.5.2 InverseofaPartitionedMatrix............................................. 43 1.5.3 Problems....................................................................... 44 1.6 Determinant,AdjugateMatrix,andCramer’sRule ............................. 47 1.6.1 AdjugateMatrix .............................................................. 53 1.6.2 Cramer’sRule................................................................. 54 1.6.3 Problems....................................................................... 57 1.7 LinearIndependence,BasisandDimension...................................... 61 1.7.1 LinearIndependence......................................................... 62 1.7.2 VectorSpacesandSubspaces............................................... 65 1.7.3 Problems....................................................................... 69 KeyTerms..................................................................................... 71 (cid:2) MATLAB Commands..................................................................... 72 MathematicaCommands .................................................................... 72 References..................................................................................... 73 2. MatrixTheory................................................................................ 75 2.1 EigenvaluesandEigenvectors ...................................................... 75 2.1.1 TheAdjugateMatrixMethodforFindinganEigenvector ............. 79 2.1.2 ComplexNumbers ........................................................... 82 2.1.3 ComplexEigenvaluesandEigenvectors.................................. 83 2.1.4 EigenvaluesandEigenvectorsofTriangular andDiagonalMatrices....................................................... 85 iii iv Contents 2.1.5 MATLAB(cid:2)andMathematicaTM............................................. 86 2.1.6 Problems....................................................................... 87 2.2 BasisofEigenvectorsandDiagonalization........................................ 91 2.2.1 DiagonalizingaMatrix ...................................................... 95 2.2.2 DeficientEigenvalues........................................................ 98 2.2.3 Problems....................................................................... 99 2.3 InnerProductandOrthogonalSetsofVectors................................... 101 2.3.1 OrthogonalSetofVectors................................................... 105 2.3.2 TheGram–SchmidtProcess................................................. 106 2.3.3 OrthogonalProjections ...................................................... 109 2.3.4 Problems....................................................................... 112 2.4 OrthonormalBasesandOrthogonalMatrices.................................... 114 2.4.1 OrthogonalSetsandBases .................................................. 114 2.4.2 OrthogonalMatrices ......................................................... 116 2.4.3 Appendix ...................................................................... 118 2.4.4 Problems....................................................................... 119 2.5 LeastSquaresSolutions.............................................................. 121 2.5.1 TheNormalEquations....................................................... 123 2.5.1.1 LeastSquaresSolutionandOrthogonalMatrices............ 127 2.5.2 Problems....................................................................... 129 2.6 SymmetricMatrices,DefiniteMatrices,andApplications...................... 131 2.6.1 ASpectralTheorem .......................................................... 131 2.6.1.1 ASpectralFormula ............................................... 135 2.6.1.2 PositiveDefiniteandPo√sitiveSemi-DefiniteMatrices ...... 136 2.6.1.3 ApplicationtoA2,A−1, A...................................... 137 2.6.1.4 ApplicationtoLeastSquaresSolutions........................ 138 2.6.2 FurtherStudyofPositiveDefiniteMatrices .............................. 139 2.6.2.1 VibrationsandtheGeneralizedEigenvalueProblem ....... 140 2.6.2.2 PositiveDefinitenessandDeterminants....................... 141 2.6.3 Problems....................................................................... 146 2.7 Factorizations:QRandSVD......................................................... 148 2.7.1 QRFactorization.............................................................. 148 2.7.2 QRandSolvingSystems..................................................... 150 2.7.3 QRandLeastSquaresSolutions............................................ 151 2.7.4 SVD............................................................................. 151 2.7.5 SVDandL.S.S. ................................................................ 158 2.7.6 Moore–PenroseGeneralizedInverse ...................................... 159 2.7.7 Problems....................................................................... 163 2.8 Factorizations:LUandCholesky................................................... 165 2.8.1 LUFactorizations............................................................. 165 2.8.2 CholeskyFactorizations ..................................................... 168 2.8.3 Problems....................................................................... 169 2.9 RayleighQuotient..................................................................... 170 2.9.1 ARayleighTheorem ......................................................... 172 2.9.2 Problems....................................................................... 174 2.10 ShortTake:InnerProductandHilbertSpaces.................................... 175 2.10.1 LinearFunctionalsandOperators ......................................... 177 2.10.2 NormandBoundedLinearOperators .................................... 179 2.10.3 Convergence,CauchyCompleteness,andHilbertSpaces ............. 183 Contents v 2.10.4 BoundedLinearFunctionalsandOperatorAdjoint..................... 186 2.10.5 ApplicationtoSignalRestoration.......................................... 187 2.10.6 ProjectionandMinimization................................................ 188 2.10.7 WeakConvergenceandCompactness .................................... 189 2.10.8 Problems....................................................................... 191 KeyTerms..................................................................................... 192 (cid:2) MATLAB Commands..................................................................... 194 MathematicaCommands .................................................................... 194 References..................................................................................... 194 3. ScalarODEsI:HomogeneousProblems................................................ 195 3.1 LinearFirst-OrderODEs............................................................. 195 3.1.1 ScalarODEs ................................................................... 195 3.1.2 LinearFirst-OrderODEs .................................................... 196 3.1.3 Steady-StateandTransientSolutions...................................... 202 3.1.4 Problems....................................................................... 205 3.2 SeparableandExactODEs .......................................................... 209 3.2.1 SeparableODEs............................................................... 209 3.2.2 ExactODEs .................................................................... 211 3.2.3 ExistenceofSolution(s)ofanIVP.......................................... 215 3.2.4 Problems....................................................................... 219 3.3 Second-OrderLinearHomogeneousODEs....................................... 222 3.3.1 Spring–Mass–DamperSystems............................................. 222 3.3.2 SeriesRLCCircuit ............................................................ 225 3.3.3 TheUnderdampedCase..................................................... 230 3.3.4 TheAmplitudeandPhaseForm ........................................... 232 3.3.5 FiguresofMeritinGraphsofUnderdampedSolutions................ 235 3.3.6 TheCriticallyDampedCase................................................ 237 3.3.7 TheWronskianDeterminant................................................ 238 3.3.8 Problems....................................................................... 239 3.4 Higher-OrderLinearODEs.......................................................... 244 3.4.1 TheZooofSolutionsofLCCHODEs ...................................... 250 3.4.2 DifferentialOperatorNotation ............................................. 251 3.4.3 ShiftTheorem ................................................................. 253 3.4.4 Problems....................................................................... 254 3.5 Cauchy–EulerODEs.................................................................. 255 3.5.1 Problems....................................................................... 260 KeyTerms..................................................................................... 261 MathematicaCommand...................................................................... 262 Reference...................................................................................... 262 4. ScalarODEsII:NonhomogeneousProblems.......................................... 263 4.1 NonhomogeneousODEs ............................................................ 263 4.1.1 SpecialCase:RHSfromtheZooandConstant CoefficientsonLHS .......................................................... 264 4.1.2 TheMethodofCoefficientstoBeDetermined ........................... 265 4.1.3 JustificationfortheMethod................................................. 269 4.1.4 UsingaShiftTheorem ....................................................... 270 4.1.5 Problems....................................................................... 271 vi Contents 4.2 ForcedOscillations.................................................................... 273 4.2.1 TheResonanceCase.......................................................... 274 4.2.2 Steady-StateSolution,FrequencyResponse,and PracticalResonance........................................................... 276 4.2.3 MaximumFrequencyResponse............................................ 281 4.2.4 BeatsPhenomenon,FastandSlowFrequencies,and FrequencyResponse.......................................................... 283 4.2.5 Problems....................................................................... 287 4.3 VariationofParameters.............................................................. 291 4.3.1 MethodofVariationofParameters........................................ 294 4.3.2 Problems....................................................................... 298 4.4 LaplaceTransforms:BasicTechniques ............................................ 299 4.4.1 Problems....................................................................... 305 4.5 LaplaceTransforms:UnitStepandOtherTechniques.......................... 307 4.5.1 WritingaFunctioninTermsofStepFunction(s) ........................ 308 4.5.2 GraphofaSolutionofanODEInvolvingaStepFunction............. 310 4.5.3 Convolution ................................................................... 312 4.5.4 ConvolutionandParticularSolutions..................................... 314 4.5.5 Delta“Functions”............................................................. 318 4.5.6 LaplaceTransformofaPeriodicFunction................................ 321 4.5.7 Remarks........................................................................ 321 4.5.8 Problems....................................................................... 321 4.6 ScalarDifferenceEquations ......................................................... 323 4.6.1 GeneralSolutionandtheCasoratiDeterminant......................... 330 4.6.2 NonhomogeneousLinearDifferenceEquation .......................... 334 4.6.3 TheMethodofUndeterminedCoefficients............................... 334 4.6.4 Problems....................................................................... 338 4.7 ShortTake:z-Transforms............................................................ 340 4.7.1 SinusoidalSignals ............................................................ 344 4.7.2 Steady-StateSolution......................................................... 345 4.7.3 Convolutionandz-Transforms............................................. 348 4.7.4 TransferFunction............................................................. 348 4.7.5 Problems....................................................................... 349 KeyTerms..................................................................................... 349 References..................................................................................... 351 5. LinearSystemsofODEs................................................................... 353 5.1 SystemsofODEs...................................................................... 353 5.1.1 SystemsofSecond-OrderEquations....................................... 357 5.1.2 CompartmentModels........................................................ 358 5.1.3 Problems....................................................................... 360 5.2 SolvingLinearHomogenousSystemsofODEs.................................. 362 5.2.1 FundamentalMatrixandetA................................................ 368 5.2.2 EquivalenceofSecond-OrderLCCHODEandLCCHSinR2.......... 372 5.2.3 MaclaurinSeriesforetA...................................................... 375 5.2.4 NonconstantCoefficients.................................................... 376 5.2.5 Problems....................................................................... 377 5.3 ComplexorDeficientEigenvalues ................................................. 381 Contents vii 5.3.1 ComplexEigenvalues........................................................ 381 5.3.2 SolvingHomogeneousSystemsofSecond-OrderEquations.......... 385 5.3.3 DeficientEigenvalues........................................................ 387 5.3.4 LaplaceTransformsand etA ................................................ 390 5.3.5 Stability......................................................................... 391 5.3.6 Problems....................................................................... 392 5.4 NonhomogeneousLinearSystems................................................. 395 5.4.1 Problems....................................................................... 401 5.5 NonresonantNonhomogeneousSystems......................................... 403 5.5.1 SinusoidalForcing............................................................ 408 5.5.2 Problems....................................................................... 411 5.6 LinearControlTheory:CompleteControllability................................ 412 5.6.1 SomeOtherControlProblems.............................................. 419 5.6.2 Problems....................................................................... 421 5.7 LinearSystemsofDifferenceEquations........................................... 422 5.7.1 ColorBlindness ............................................................... 423 5.7.2 GeneralSolutionandtheCasoratiDeterminant......................... 425 5.7.3 ComplexEigenvalues........................................................ 427 5.7.4 EquivalenceofSecond-OrderScalarDifferenceEquation and aSysteminR2........................................................... 428 5.7.5 LadderNetworkElectricalCircuits........................................ 429 5.7.6 Stability......................................................................... 434 5.7.7 Problems....................................................................... 436 5.8 ShortTake:PeriodicLinearDifferentialEquations.............................. 439 5.8.1 TheStroboscopic,or“Return,”Map....................................... 441 5.8.2 FloquetRepresentation ...................................................... 442 5.8.3 Stability......................................................................... 445 5.8.4 Hill’sEquation ................................................................ 447 5.8.5 PeriodicSolutionofaNonhomogeneous ODESystem ................................................................... 448 5.8.6 Problems....................................................................... 451 KeyTerms..................................................................................... 454 (cid:2) MATLAB Commands..................................................................... 455 References..................................................................................... 455 6. Geometry,Calculus,andOtherTools................................................... 457 6.1 DotProduct,CrossProduct,Lines,andPlanes................................... 457 6.1.1 DotProductandCrossProduct ............................................ 457 6.1.2 Lines............................................................................ 459 6.1.3 Planes........................................................................... 459 6.1.4 Problems....................................................................... 461 6.2 Trigonometry,Polar,Cylindrical,and SphericalCoordinates................................................................ 463 6.2.1 CylindricalCoordinates ..................................................... 465 6.2.2 SphericalCoordinates........................................................ 465 6.2.3 Right-HandedOrthogonalBasesforR3................................... 467 6.2.4 OrthonormalBasisinSphericalCoordinates............................. 468 6.2.5 RelationshipstotheStandardo.n.Basis .................................. 470 6.2.6 Problems....................................................................... 471 viii Contents 6.3 CurvesandSurfaces.................................................................. 471 6.3.1 CurvesandCalculus......................................................... 474 6.3.2 ZhukovskiiAirfoil............................................................ 477 6.3.3 Surfaces ........................................................................ 478 6.3.4 Problems....................................................................... 482 6.4 PartialDerivatives .................................................................... 485 6.4.1 LinearApproximation....................................................... 486 6.4.2 MultivariableChainRules .................................................. 489 6.4.3 GradientVectorinR3 ........................................................ 492 6.4.4 ScalarPotentialFunctions................................................... 493 6.4.5 Problems....................................................................... 495 6.5 TangentPlaneandNormalVector................................................. 498 6.5.1 Problems....................................................................... 504 6.6 Area,Volume,andLinearTransformations ...................................... 505 6.6.1 LinearTransformations...................................................... 511 6.6.2 LinearTransformations,Area,andVolume.............................. 514 6.6.3 ChangeofVariables,Area,andVolume.................................. 516 6.6.4 ElementofSurfaceArea..................................................... 519 6.6.5 Problems....................................................................... 520 6.7 DifferentialOperatorsandCurvilinearCoordinates............................ 522 6.7.1 PropertiesoftheOperatorsgrad,div,andcurl........................... 524 6.7.2 CurvilinearCoordinates..................................................... 525 6.7.3 DifferentialOperatorsinCurvilinearCoordinates...................... 529 6.7.4 SummaryofOperatorsinCylindricalCoordinates ..................... 533 6.7.5 SummaryofOperatorsinSphericalCoordinates........................ 533 6.7.6 Problems....................................................................... 534 6.8 RotatingCoordinateFrames ........................................................ 537 6.8.1 ODEsDescribingRotation .................................................. 537 6.8.2 VelocityandAcceleration................................................... 540 6.8.3 VelocityandAccelerationinaRotatingFrameWhose OriginIsMoving.............................................................. 543 6.8.4 Problems....................................................................... 544 KeyTerms..................................................................................... 546 MathematicaCommand...................................................................... 548 Reference...................................................................................... 548 7. IntegralTheorems,MultipleIntegrals,andApplications........................... 549 7.1 IntegralsforaFunctionofaSingleVariable...................................... 549 7.1.1 ImproperIntegrals............................................................ 553 7.1.2 Problems....................................................................... 554 7.2 LineIntegrals.......................................................................... 555 7.2.1 LineIntegralsofVector-ValuedFunctions ............................... 560 7.2.2 FundamentalTheoremofLineIntegrals.................................. 563 7.2.3 PathDirection................................................................. 565 7.2.4 OtherNotations............................................................... 565 7.2.5 Problems....................................................................... 567 7.3 DoubleIntegrals,Green’sTheorem,andApplications.......................... 570 7.3.1 DoubleIntegralasVolume.................................................. 574 7.3.2 PolarCoordinates............................................................. 580