Advanced Engineering Mathematics

1459 Pages·2013·16.197 MB·English

by  Lawrence Turyn

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
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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.
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Version Date: 20130801
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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