Higher Engineering Mathematics

924 Pages·2017·14.15 MB·English

by  John Bird

Higher Engineering Mathematics
Nowinits eighthedition,HigherEngineeringMathe- JohnBird,BSc(Hons),CMath,CEng,CSci,FITE,
maticshashelpedthousandsofstudentssucceedintheir FIMA,FCollT,istheformerHeadofAppliedElectron-
exams.Theoryiskepttoaminimum,withtheemphasis ics intheFacultyofTechnologyatHighburyCollege,
firmlyplacedonproblem-solvingskills,makingthisa Portsmouth,UK.Morerecentlyhehascombinedfree-
thoroughlypracticalintroductiontotheadvancedengi- lance lecturing and examining, and is the author of
neeringmathematicsthatstudentsneedtomaster.The over 130 textbooks on engineering and mathemati-
extensive and thorough topic coverage makes this an cal subjects with worldwide sales of over one mil-
ideal text for upper-level vocational courses and for lion copies. He is currently lecturing at the Defence
undergraduate degree courses. It is also supported by School of Marine Engineering in the Defence Col-
a fullyupdatedcompanionwebsite with resourcesfor lege of Technical Training at HMS Sultan, Gosport,
both students and lecturers. It has full solutions to all Hampshire,UK.
2,000 further questions contained in the 277 practice
exercises.
Why is knowledge of mathematics important in engineering?
A career in any engineering or scientific field will Aerospace engineers require mathematics to perform
requirebothbasicandadvancedmathematics.Without a varietyofengineeringworkin designing,construct-
mathematicstodetermineprinciples,calculatedimen- ing, andtestingaircraft,missiles, andspacecraft;they
sionsandlimits,explorevariations,proveconcepts,and conduct basic and applied research to evaluate adapt-
soon,therewouldbenomobiletelephones,televisions, abilityofmaterialsandequipmenttoaircraftdesignand
stereosystems,videogames,microwaveovens,comput- manufactureand recommendimprovementsin testing
ers,orvirtuallyanythingelectronic.Therewouldbeno equipmentandtechniques.
bridges,tunnels,roads,skyscrapers,automobiles,ships,
Nuclear engineers require mathematics to conduct
planes,rocketsormostthingsmechanical.Therewould
researchonnuclearengineeringproblemsorapplyprin-
be no metals beyond the common ones, such as iron
ciples and theory of nuclear science to problemscon-
and copper,no plastics, no synthetics. In fact, society
cernedwith release,control,andutilisationofnuclear
wouldmostcertainlybelessadvancedwithouttheuse
energyandnuclearwastedisposal.
of mathematics throughout the centuries and into the
future. Petroleum engineers require mathematics to devise
methods to improve oil and gas well production and
Electrical engineers require mathematics to design,
determine the need for new or modified tool designs;
develop,test,orsupervisethemanufacturingandinstal-
they oversee drilling and offer technical advice to
lationofelectricalequipment,components,orsystems
forcommercial,industrial,military,orscientificuse. achieveeconomicalandsatisfactoryprogress.
Mechanicalengineersrequiremathematicstoperform Industrial engineers require mathematics to design,
engineering duties in planning and designing tools, develop,test,andevaluateintegratedsystemsforman-
engines,machines,andothermechanicallyfunctioning agingindustrialproductionprocesses,includinghuman
equipment;theyoverseeinstallation,operation,mainte- work factors, qualitycontrol,inventorycontrol,logis-
nance,andrepairofsuchequipmentascentralisedheat, tics and material flow, cost analysis, and production
gas,water,andsteamsystems. co-ordination.
Environmental engineers require mathematics to mathematicaltoolssuchasdifferentialequations,tensor
design, plan, or perform engineering duties in the analysis,fieldtheory,numericalmethodsandoperations
prevention, control, and remediation of environmen- research.
tal health hazards, using various engineering disci-
Knowledgeofmathematicsisthereforeneededbyeach
plines; their work may include waste treatment, site
oftheengineeringdisciplineslistedabove.
remediation,orpollutioncontroltechnology.
Itisintendedthatthistext–HigherEngineeringMathe-
Civil engineers require mathematics in all levels in matics–willprovideastep-by-stepapproachtolearning
civil engineering – structural engineering, hydraulics fundamentalmathematicsneededforyourengineering
andgeotechnicalengineeringareallfieldsthatemploy studies.
Higher Engineering Mathematics
Eighth Edition
John Bird
Eightheditionpublished2017
byRoutledge
2ParkSquare,MiltonPark,Abingdon,OxonOX144RN
andbyRoutledge
711ThirdAvenue,NewYork,NY10017
RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness
©2017JohnBird
TherightofJohnBirdtobeidentifiedasauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78
oftheCopyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyformorbyanyelectronic,mechanical,
orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor
retrievalsystem,withoutpermissioninwritingfromthepublishers.
Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentification
andexplanationwithoutintenttoinfringe.
FirsteditionpublishedbyElsevier1993
SeventheditionpublishedbyRoutledge2014
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationData
Names:Bird,J.O.Title:Higherengineeringmathematics/JohnBird.
Description:8thed.|BocaRaton:CRCPress,2017.|Includesindex.
Identifiers:LCCN2016038671|ISBN9781138673571
Subjects:LCSH:Engineeringmathematics.
Classification:LCCTA330.B522017|DDC620.001/51–dc23LC
recordavailableathttps://lccn.loc.gov/2016038671
ISBN:978-1-138-67357-1(pbk)
ISBN:978-1-315-26502-5(ebk)
TypesetinTimesby
ServisFilmsettingLtd,Stockport,Cheshire
Visitthecompanionwebsite:www.routledge.com/cw/bird
To Sue
Contents
Preface xiii
RevisionTest1 49
Syllabusguidance xv
6 Arithmeticandgeometricprogressions 50
6.1 Arithmeticprogressions 50
SectionA Numberandalgebra 1
6.2 Workedproblemsonarithmeticprogressions 51
6.3 Furtherworkedproblemsonarithmetic
1 Algebra 3 progressions 52
1.1 Introduction 3
6.4 Geometricprogressions 53
1.2 Revisionofbasiclaws 3
6.5 Workedproblemsongeometricprogressions 54
1.3 Revisionofequations 5
6.6 Furtherworkedproblemsongeometric
1.4 Polynomialdivision 8 progressions 55
1.5 Thefactortheorem 10
7 Thebinomialseries 58
1.6 Theremaindertheorem 12
7.1 Pascal’striangle 58
2 Partialfractions 15 7.2 Thebinomialseries 60
2.1 Introductiontopartialfractions 15 7.3 Workedproblemsonthebinomialseries 60
2.2 Workedproblemsonpartialfractionswith 7.4 Furtherworkedproblemsonthebinomial
linearfactors 16 series 62
2.3 Workedproblemsonpartialfractionswith 7.5 Practicalproblemsinvolvingthebinomial
repeatedlinearfactors 18 theorem 64
2.4 Workedproblemsonpartialfractionswith
8 Maclaurin’sseries 68
quadraticfactors 20
8.1 Introduction 69
3 Logarithms 22 8.2 DerivationofMaclaurin’stheorem 69
3.1 Introductiontologarithms 22 8.3 ConditionsofMaclaurin’sseries 70
3.2 Lawsoflogarithms 24 8.4 WorkedproblemsonMaclaurin’sseries 70
3.3 Indicialequations 27 8.5 NumericalintegrationusingMaclaurin’s
3.4 Graphsoflogarithmicfunctions 28 series 73
8.6 Limitingvalues 75
4 Exponentialfunctions 29
4.1 Introductiontoexponentialfunctions 29
4.2 Thepowerseriesforex 30 RevisionTest2 78
4.3 Graphsofexponentialfunctions 32
4.4 Napierianlogarithms 33 9 Solvingequationsbyiterativemethods 79
9.1 Introductiontoiterativemethods 79
4.5 Lawsofgrowthanddecay 36
9.2 Thebisectionmethod 80
4.6 Reductionofexponentiallawstolinearform 40
9.3 Analgebraicmethodofsuccessive
5 Inequalities 43 approximations 83
5.1 Introductiontoinequalities 43 9.4 TheNewton-Raphsonmethod 86
5.2 Simpleinequalities 44
10 Binary,octalandhexadecimalnumbers 90
5.3 Inequalitiesinvolvingamodulus 44
10.1 Introduction 90
5.4 Inequalitiesinvolvingquotients 45
10.2 Binarynumbers 91
5.5 Inequalitiesinvolvingsquarefunctions 46
10.3 Octalnumbers 94
5.6 Quadraticinequalities 47
10.4 Hexadecimalnumbers 96
vii
Contents
11 Booleanalgebraandlogiccircuits 100 15.2 Anglesofanymagnitude 163
11.1 Booleanalgebraandswitchingcircuits 101 15.3 Theproductionofasineandcosinewave 166
11.2 SimplifyingBooleanexpressions 105 15.4 Sineandcosinecurves 167
11.3 LawsandrulesofBooleanalgebra 105 15.5 SinusoidalformAsin(ωt α) 171
±
11.4 DeMorgan’slaws 107 15.6 Harmonicsynthesiswithcomplex
11.5 Karnaughmaps 108 waveforms 174
11.6 Logiccircuits 112
16 Hyperbolicfunctions 180
11.7 Universallogicgates 116
16.1 Introductiontohyperbolicfunctions 180
16.2 Graphsofhyperbolicfunctions 182
RevisionTest3 119
16.3 Hyperbolicidentities 184
16.4 Solvingequationsinvolvinghyperbolic
functions 186
SectionB Geometryandtrigonometry 121
16.5 Seriesexpansionsforcoshxandsinhx 188
17 Trigonometricidentitiesandequations 190
12 Introductiontotrigonometry 123 17.1 Trigonometricidentities 190
12.1 Trigonometry 124 17.2 Workedproblemsontrigonometric
12.2 ThetheoremofPythagoras 124 identities 191
12.3 Trigonometricratiosofacuteangles 125 17.3 Trigonometricequations 192
12.4 Evaluatingtrigonometricratios 127 17.4 Workedproblems(i)ontrigonometric
12.5 Solutionofright-angledtriangles 131 equations 193
12.6 Anglesofelevationanddepression 133 17.5 Workedproblems(ii)ontrigonometric
12.7 Sineandcosinerules 134 equations 194
12.8 Areaofanytriangle 135 17.6 Workedproblems(iii)ontrigonometric
equations 195
12.9 Workedproblemsonthesolutionof
trianglesandfindingtheirareas 135 17.7 Workedproblems(iv)ontrigonometric
equations 195
12.10 Furtherworkedproblemsonthesolution
oftrianglesandfindingtheirareas 136
18 Therelationshipbetweentrigonometricand
12.11 Practicalsituationsinvolvingtrigonometry 138
hyperbolicfunctions 198
12.12 Furtherpracticalsituationsinvolving 18.1 Therelationshipbetweentrigonometric
trigonometry 140 andhyperbolicfunctions 198
18.2 Hyperbolicidentities 199
13 Cartesianandpolarco-ordinates 143
13.1 Introduction 144 19 Compoundangles 202
13.2 ChangingfromCartesianintopolar 19.1 Compoundangleformulae 202
co-ordinates 144 19.2 Conversionofasinωt+bcosωtinto
13.3 ChangingfrompolarintoCartesian Rsin(ωt+α) 204
co-ordinates 146 19.3 Doubleangles 208
13.4 UseofPol/Recfunctionsoncalculators 147 19.4 Changingproductsofsinesandcosines
intosumsordifferences 210
14 Thecircleanditsproperties 149
19.5 Changingsumsordifferencesofsinesand
14.1 Introduction 149
cosinesintoproducts 211
14.2 Propertiesofcircles 149
19.6 Powerwaveformsina.c.circuits 212
14.3 Radiansanddegrees 151
14.4 Arclengthandareaofcirclesandsectors 152
RevisionTest5 216
14.5 Theequationofacircle 155
14.6 Linearandangularvelocity 156
14.7 Centripetalforce 158 SectionC Graphs 217
RevisionTest4 160
20 Functionsandtheircurves 219
20.1 Standardcurves 219
15 Trigonometricwaveforms 162 20.2 Simpletransformations 222
15.1 Graphsoftrigonometricfunctions 162 20.3 Periodicfunctions 227
viii
Contents
20.4 Continuousanddiscontinuousfunctions 227 25.2 Solutionofsimultaneousequationsby
20.5 Evenandoddfunctions 228 determinants 290
20.6 Inversefunctions 229 25.3 Solutionofsimultaneousequationsusing
20.7 Asymptotes 231 Cramer’srule 293
20.8 Briefguidetocurvesketching 237 25.4 Solutionofsimultaneousequationsusing
theGaussianeliminationmethod 294
20.9 Workedproblemsoncurvesketching 238
25.5 Eigenvaluesandeigenvectors 296
21 Irregularareas,volumesandmeanvaluesof
waveforms 241 RevisionTest7 302
21.1 Areasofirregularfigures 241
21.2 Volumesofirregularfigures 244
21.3 Themeanoraveragevalueofawaveform 245 SectionF Vectorgeometry 303
RevisionTest6 250
26 Vectors 305
26.1 Introduction 305
26.2 Scalarsandvectors 305
SectionD Complexnumbers 251
26.3 Drawingavector 306
26.4 Additionofvectorsbydrawing 307
22 Complexnumbers 253 26.5 Resolvingvectorsintohorizontaland
22.1 Cartesiancomplexnumbers 254 verticalcomponents 309
22.2 TheArganddiagram 255 26.6 Additionofvectorsbycalculation 310
22.3 Additionandsubtractionofcomplex 26.7 Vectorsubtraction 314
numbers 255 26.8 Relativevelocity 316
22.4 Multiplicationanddivisionofcomplex 26.9 i,j andknotation 317
numbers 256
22.5 Complexequations 258 27 Methodsofaddingalternatingwaveforms 319
22.6 Thepolarformofacomplexnumber 259 27.1 Combinationoftwoperiodicfunctions 319
22.7 Multiplicationanddivisioninpolarform 261 27.2 Plottingperiodicfunctions 320
22.8 Applicationsofcomplexnumbers 262 27.3 Determiningresultantphasorsby
drawing 321
23 DeMoivre’stheorem 266 27.4 Determiningresultantphasorsbythesine
23.1 Introduction 267 andcosinerules 323
23.2 Powersofcomplexnumbers 267 27.5 Determiningresultantphasorsby
23.3 Rootsofcomplexnumbers 268 horizontalandverticalcomponents 324
23.4 Theexponentialformofacomplexnumber 270 27.6 Determiningresultantphasorsbyusing
complexnumbers 326
23.5 Introductiontolocusproblems 271
28 Scalarandvectorproducts 330
SectionE Matricesanddeterminants 275 28.1 Theunittriad 330
28.2 Thescalarproductoftwovectors 331
28.3 Vectorproducts 335
24 Thetheoryofmatricesanddeterminants 277
28.4 Vectorequationofaline 339
24.1 Matrixnotation 277
24.2 Addition,subtractionandmultiplication
RevisionTest8 341
ofmatrices 278
24.3 Theunitmatrix 281
24.4 Thedeterminantofa2by2matrix 281 SectionG Introductiontocalculus 343
24.5 Theinverseorreciprocalofa2by2matrix 282
24.6 Thedeterminantofa3by3matrix 283
24.7 Theinverseorreciprocalofa3by3matrix 285 29 Methodsofdifferentiation 345
29.1 Introductiontocalculus 345
25 Applicationsofmatricesanddeterminants 287 29.2 Thegradientofacurve 345
25.1 Solutionofsimultaneousequationsby 29.3 Differentiationfromfirstprinciples 346
matrices 288 29.4 Differentiationofcommonfunctions 347
ix
Contents
29.5 Differentiationofaproduct 350 34.3 Differentiationinparameters 414
29.6 Differentiationofaquotient 352 34.4 Furtherworkedproblemson
29.7 Functionofafunction 353 differentiationofparametricequations 416
29.8 Successivedifferentiation 355
35 Differentiationofimplicitfunctions 419
30 Someapplicationsofdifferentiation 357 35.1 Implicitfunctions 419
30.1 Ratesofchange 357 35.2 Differentiatingimplicitfunctions 419
30.2 Velocityandacceleration 359 35.3 Differentiatingimplicitfunctions
30.3 Turningpoints 362 containingproductsandquotients 420
30.4 Practicalproblemsinvolvingmaximum 35.4 Furtherimplicitdifferentiation 421
andminimumvalues 365
30.5 Pointsofinflexion 369 36 Logarithmicdifferentiation 425
36.1 Introductiontologarithmic
30.6 Tangentsandnormals 371
differentiation 425
30.7 Smallchanges 372
36.2 Lawsoflogarithms 425
36.3 Differentiationoflogarithmicfunctions 426
RevisionTest9 375
36.4 Differentiationoffurtherlogarithmic
functions 426
31 Standardintegration 376 36.5 Differentiationof[f(x)]x 428
31.1 Theprocessofintegration 376
31.2 Thegeneralsolutionofintegralsofthe
formaxn 377 RevisionTest11 431
31.3 Standardintegrals 377
31.4 Definiteintegrals 380 37 Differentiationofhyperbolicfunctions 432
37.1 Standarddifferentialcoefficientsof
32 Someapplicationsofintegration 383
hyperbolicfunctions 432
32.1 Introduction 384
37.2 Furtherworkedproblemson
32.2 Areasunderandbetweencurves 384
differentiationofhyperbolicfunctions 433
32.3 Meanandrmsvalues 385
32.4 Volumesofsolidsofrevolution 386 38 Differentiationofinversetrigonometricand
32.5 Centroids 388 hyperbolicfunctions 435
32.6 TheoremofPappus 390 38.1 Inversefunctions 435
32.7 Secondmomentsofareaofregularsections 392 38.2 Differentiationofinversetrigonometric
functions 437
33 Introductiontodifferentialequations 400 38.3 Logarithmicformsoftheinverse
33.1 Familyofcurves 400 hyperbolicfunctions 440
33.2 Differentialequations 401 38.4 Differentiationofinversehyperbolic
33.3 Thesolutionofequationsoftheform functions 442
dy
f(x) 402
dx = 39 Partialdifferentiation 446
33.4 Thesolutionofequationsoftheform
39.1 Introductiontopartialdifferentiation 446
dy
f(y) 403 39.2 Firstorderpartialderivatives 446
dx =
33.5 Thesolutionofequationsoftheform 39.3 Secondorderpartialderivatives 449
dy
f(x).f(y) 405
40 Totaldifferential,ratesofchangeandsmall
dx =
changes 452
40.1 Totaldifferential 452
RevisionTest10 409
40.2 Ratesofchange 453
40.3 Smallchanges 456
SectionH Furtherdifferentialcalculus 411
41 Maxima,minimaandsaddlepointsforfunctions
oftwovariables 459
34 Differentiationofparametricequations 413 41.1 Functionsoftwoindependent
34.1 Introductiontoparametricequations 413 variables 459
34.2 Somecommonparametricequations 414 41.2 Maxima,minimaandsaddlepoints 460