Table Of ContentBird’s Higher Engineering Mathematics
Whyisknowledgeofmathematicsimportantinengineering?
A career in any engineering or scientific field will principles and theory of nuclear science to prob-
requirebothbasicandadvancedmathematics.Without lemsconcernedwithrelease,controlandutilisationof
mathematics to determine principles, calculate dimen- nuclearenergyandnuclearwastedisposal.
sionsandlimits,explorevariations,proveconceptsand
Petroleum engineers require mathematics to devise
soon,therewouldbenomobiletelephones,televisions,
methods to improve oil and gas well production and
stereo systems, video games, microwave ovens, com-
determine the need for new or modified tool designs;
putersorvirtuallyanythingelectronic.Therewouldbe
they oversee drilling and offer technical advice to
no bridges, tunnels, roads, skyscrapers, automobiles,
achieveeconomicalandsatisfactoryprogress.
ships,planes,rocketsormostthingsmechanical.There
wouldbenometalsbeyondthecommonones,suchas Industrial engineers require mathematics to design,
ironandcopper,noplastics,nosynthetics.Infact,soci- develop,testandevaluateintegratedsystemsforman-
etywouldmostcertainlybelessadvancedwithoutthe agingindustrialproductionprocesses,includinghuman
use of mathematics throughout the centuries and into work factors, quality control, inventory control, logis-
thefuture. tics and material flow, cost analysis and production
co-ordination.
Electrical engineers require mathematics to design,
develop,testorsupervisethemanufacturingandinstal- Environmental engineers require mathematics to
lation of electrical equipment, components or systems design, plan or perform engineering duties in the
forcommercial,industrial,militaryorscientificuse. prevention, control and remediation of environmen-
tal health hazards, using various engineering disci-
Mechanicalengineersrequiremathematicstoperform
plines; their work may include waste treatment, site
engineering duties in planning and designing tools,
remediationorpollutioncontroltechnology.
engines,machinesandothermechanicallyfunctioning
equipment;theyoverseeinstallation,operation,mainte- Civil engineers require mathematics in all levels in
nanceandrepairofsuchequipmentascentralisedheat, civil engineering – structural engineering, hydraulics
gas,waterandsteamsystems. andgeotechnicalengineeringareallfieldsthatemploy
mathematical tools such as differential equations, ten-
Aerospace engineers require mathematics to perform
soranalysis,fieldtheory,numericalmethodsandoper-
a variety of engineering work in designing, construct-
ationsresearch.
ing and testing aircraft, missiles and spacecraft; they
conduct basic and applied research to evaluate adapt- Knowledgeofmathematicsisthereforeneededbyeach
abilityofmaterialsandequipmenttoaircraftdesignand oftheengineeringdisciplineslistedabove.
manufacture and recommend improvements in testing
It is intended that this text – Bird’s Higher Engineer-
equipmentandtechniques.
ingMathematics–willprovideastep-by-stepapproach
Nuclear engineers require mathematics to conduct to learning the essential mathematics needed for your
research on nuclear engineering problems or apply engineeringstudies.
Now in its ninth edition, Bird’s Higher Engineering Mathematics has helped thousands of students to succeed in
their exams. Mathematical theories are explained in a straightforward manner, supported by practical engineering
examples and applications to ensure that readers can relate theory to practice. Some 1,200 engineering situation-
s/problemshavebeen‘flagged-up’tohelpdemonstratethatengineeringcannotbefullyunderstoodwithoutagood
knowledgeofmathematics.
The extensive and thorough topic coverage makes this an ideal text for undergraduate degree courses, founda-
tion degrees, and for higher-level vocational courses such as Higher National Certificate and Diploma courses in
engineeringdisciplines.
Itscompanionwebsiteatwww.routledge.com/cw/birdprovidesresourcesforbothstudentsandlecturers,includ-
ingfullsolutionsforall2,100furtherquestions,listsofessentialformulae,multiple-choicetests,andillustrations,
aswellasfullsolutionstorevisiontestsforcourseinstructors.
JohnBird,BSc(Hons),CEng,CMath,CSci,FIMA,FIET,FCollT,istheformerHeadofAppliedElectronicsinthe
FacultyofTechnologyatHighburyCollege,Portsmouth,UK.Morerecently,hehascombinedfreelancelecturingat
theUniversityofPortsmouth,withExaminerresponsibilitiesforAdvancedMathematicswithCityandGuildsand
examiningfortheInternationalBaccalaureateOrganisation.Hehasover45years’experienceofsuccessfullyteach-
ing,lecturing,instructing,training,educatingandplanningtraineeengineersstudyprogrammes.Heistheauthorof
146textbooksonengineering,scienceandmathematicalsubjects,withworldwidesalesofoveronemillioncopies.
Heisacharteredengineer,acharteredmathematician,acharteredscientistandaFellowofthreeprofessionalinsti-
tutions.He hasrecently retiredfrom lecturingat theRoyalNavy’sDefence Collegeof MarineEngineering inthe
DefenceCollegeofTechnicalTrainingatH.M.S.Sultan,Gosport,Hampshire,UK,oneofthelargestengineering
trainingestablishmentsinEurope.
Bird’s Higher Engineering Mathematics
Ninth Edition
John Bird
Nintheditionpublished2021
byRoutledge
2ParkSquare,MiltonPark,Abingdon,OxonOX144RN
andbyRoutledge
52VanderbiltAvenue,NewYork,NY10017
RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness
©2021JohnBird
TherightofJohnBirdtobeidentifiedasauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78
oftheCopyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyformorbyanyelectronic,mechanical,
orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor
retrievalsystem,withoutpermissioninwritingfromthepublishers.
Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentification
andexplanationwithoutintenttoinfringe.
FirsteditionpublishedbyElsevier1993
EightheditionpublishedbyRoutledge2017
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationData
Names:Bird,J.O.,author.
Title:Bird’shigherengineeringmathematics/JohnBird.
Othertitles:Higherengineeringmathematics
Description:Ninthedition.|Abingdon,Oxon;NewYork:Routledge,2021.
|Includesindex.
Identifiers:LCCN2021000158(print)|LCCN2021000159(ebook)|ISBN
9780367643737(paperback)|ISBN9780367643751(hardback)|ISBN
9781003124221(ebook)
Subjects:LCSH:Engineeringmathematics.
Classification:LCCTA330.B522021(print)|LCCTA330(ebook)|DDC
620.001/51–dc23
LCrecordavailableathttps://lccn.loc.gov/2021000158
LCebookrecordavailableathttps://lccn.loc.gov/2021000159
ISBN:978-0-367-64375-1(hbk)
ISBN:978-0-367-64373-7(pbk)
ISBN:978-1-003-12422-1(ebk)
TypesetinTimes
byKnowledgeWorksGlobalLtd.
Visitthecompanionwebsite:www.routledge.com/cw/bird
To Sue
Contents
Preface xiv 5.4 Furtherworkedproblemsonthe
binomialseries 51
Syllabusguidance xvi 5.5 Practicalproblemsinvolvingthe
binomialtheorem 54
SectionA Numberandalgebra 1 6 Solvingequationsbyiterativemethods 57
6.1 Introductiontoiterativemethods 57
6.2 Thebisectionmethod 58
1 Algebra 3
6.3 Analgebraicmethodofsuccessive
1.1 Introduction 3
approximations 61
1.2 Revisionofbasiclaws 3
1.3 Revisionofequations 5 7 Booleanalgebraandlogiccircuits 65
1.4 Polynomialdivision 9 7.1 Booleanalgebraandswitchingcircuits 66
1.5 Thefactortheorem 11 7.2 SimplifyingBooleanexpressions 70
1.6 Theremaindertheorem 13 7.3 LawsandrulesofBooleanalgebra 70
7.4 DeMorgan’slaws 72
2 Partialfractions 17 7.5 Karnaughmaps 73
2.1 Introductiontopartialfractions 17
7.6 Logiccircuits 77
2.2 Partialfractionswithlinearfactors 17
7.7 Universallogicgates 81
2.3 Partialfractionswithrepeatedlinearfactors 20
2.4 Partialfractionswithquadraticfactors 22
RevisionTest2 85
3 Logarithms 24
3.1 Introductiontologarithms 24
3.2 Lawsoflogarithms 26
SectionB Geometryandtrigonometry 87
3.3 Indicialequations 28
3.4 Graphsoflogarithmicfunctions 30
8 Introductiontotrigonometry 89
4 Exponentialfunctions 32 8.1 Trigonometry 90
4.1 Introductiontoexponentialfunctions 32 8.2 ThetheoremofPythagoras 90
4.2 Thepowerseriesforex 33 8.3 Trigonometricratiosofacuteangles 91
4.3 Graphsofexponentialfunctions 35 8.4 Evaluatingtrigonometricratios 93
4.4 Napierianlogarithms 36 8.5 Solutionofright-angledtriangles 97
4.5 Lawsofgrowthanddecay 40 8.6 Anglesofelevationanddepression 99
4.6 Reductionofexponentiallawstolinearform43 8.7 Sineandcosinerules 100
8.8 Areaofanytriangle 101
8.9 Workedproblemsonthesolutionof
RevisionTest1 47
trianglesandfindingtheirareas 101
8.10 Furtherworkedproblemsonsolving
trianglesandfindingtheirareas 102
5 Thebinomialseries 48
8.11 Practicalsituationsinvolvingtrigonometry 104
5.1 Pascal’striangle 48
8.12 Furtherpracticalsituationsinvolving
5.2 Thebinomialseries 50
trigonometry 106
5.3 Workedproblemsonthebinomialseries 50
Contents vii
9 Cartesianandpolarco-ordinates 111 14 Therelationshipbetweentrigonometricand
9.1 Introduction 112 hyperbolicfunctions 169
9.2 ChangingfromCartesianintopolar 14.1 Therelationshipbetweentrigonometric
co-ordinates 112 andhyperbolicfunctions 169
9.3 ChangingfrompolarintoCartesian 14.2 Hyperbolicidentities 170
co-ordinates 114
9.4 UseofPol/Recfunctionsoncalculators 115
15 Compoundangles 173
15.1 Compoundangleformulae 173
10 Thecircleanditsproperties 117 15.2 Conversionofasin!t+bcos!tinto
10.1 Introduction 117 Rsin(!t+(cid:11)) 175
10.2 Propertiesofcircles 117 15.3 Doubleangles 179
10.3 Radiansanddegrees 119 15.4 Changingproductsofsinesandcosines
10.4 Arclengthandareaofcirclesandsectors 120 intosumsordifferences 181
10.5 Theequationofacircle 123 15.5 Changingsumsordifferencesofsines
andcosinesintoproducts 182
10.6 Linearandangularvelocity 125
15.6 Powerwaveformsina.c.circuits 183
10.7 Centripetalforce 126
RevisionTest4 187
RevisionTest3 130
SectionC Graphs 189
11 Trigonometricwaveforms 132
11.1 Graphsoftrigonometricfunctions 132
11.2 Anglesofanymagnitude 133 16 Functionsandtheircurves 191
11.3 Theproductionofasineandcosinewave 136 16.1 Standardcurves 191
11.4 Sineandcosinecurves 137 16.2 Simpletransformations 194
11.5 SinusoidalformAsin(!t(cid:6)(cid:11)) 141 16.3 Periodicfunctions 199
11.6 Harmonicsynthesiswithcomplex 16.4 Continuousanddiscontinuousfunctions 199
waveforms 144 16.5 Evenandoddfunctions 200
16.6 Inversefunctions 201
16.7 Asymptotes 203
12 Hyperbolicfunctions 151
16.8 Briefguidetocurvesketching 208
12.1 Introductiontohyperbolicfunctions 151
16.9 Workedproblemsoncurvesketching 208
12.2 Graphsofhyperbolicfunctions 153
12.3 Hyperbolicidentities 155
12.4 Solvingequationsinvolvinghyperbolic 17 Irregularareas,volumesandmeanvaluesof
functions 157 waveforms 213
12.5 Seriesexpansionsforcoshxandsinhx 159 17.1 Areasofirregularfigures 213
17.2 Volumesofirregularsolids 216
17.3 Themeanoraveragevalueofawaveform 217
13 Trigonometricidentitiesandequations 161
13.1 Trigonometricidentities 161
13.2 Workedproblemsontrigonometric RevisionTest5 223
identities 162
13.3 Trigonometricequations 163
13.4 Workedproblems(i)ontrigonometric SectionD Complexnumbers 225
equations 164
13.5 Workedproblems(ii)ontrigonometric
equations 165 18 Complexnumbers 227
13.6 Workedproblems(iii)ontrigonometric 18.1 Cartesiancomplexnumbers 228
equations 166 18.2 TheArganddiagram 229
13.7 Workedproblems(iv)ontrigonometric 18.3 Additionandsubtractionofcomplex
equations 166 numbers 229
viii Contents
18.4 Multiplicationanddivisionofcomplex 22.5 Resolvingvectorsintohorizontaland
numbers 230 verticalcomponents 289
18.5 Complexequations 232 22.6 Additionofvectorsbycalculation 290
18.6 Thepolarformofacomplexnumber 233 22.7 Vectorsubtraction 295
18.7 Multiplicationanddivisioninpolarform 235 22.8 Relativevelocity 297
18.8 Applicationsofcomplexnumbers 236 22.9 i,jandknotation 298
23 Methodsofaddingalternatingwaveforms 301
19 DeMoivre’stheorem 241 23.1 Combinationoftwoperiodicfunctions 301
19.1 Introduction 242
23.2 Plottingperiodicfunctions 302
19.2 Powersofcomplexnumbers 242
23.3 Determiningresultantphasorsby
19.3 Rootsofcomplexnumbers 243 drawing 303
19.4 Theexponentialformofacomplexnumber 245 23.4 Determiningresultantphasorsbythe
19.5 Introductiontolocusproblems 246 sineandcosinerules 305
23.5 Determiningresultantphasorsby
horizontalandverticalcomponents 306
SectionE Matricesanddeterminants251
23.6 Determiningresultantphasorsbyusing
complexnumbers 308
20 Thetheoryofmatricesanddeterminants 253
24 Scalarandvectorproducts 312
20.1 Matrixnotation 253
24.1 Theunittriad 312
20.2 Addition,subtractionandmultiplication
24.2 Thescalarproductoftwovectors 313
ofmatrices 254
24.3 Vectorproducts 317
20.3 Theunitmatrix 257
24.4 Vectorequationofaline 321
20.4 Thedeterminantofa2by2matrix 257
20.5 Theinverseorreciprocalofa2by2matrix 258
RevisionTest7 323
20.6 Thedeterminantofa3by3matrix 259
20.7 Theinverseorreciprocalofa3by3matrix 261
SectionG Differentialcalculus 325
21 Applicationsofmatricesanddeterminants 265
21.1 Solutionofsimultaneousequationsby
25 Methodsofdifferentiation 327
matrices 266
25.1 Introductiontocalculus 327
21.2 Solutionofsimultaneousequationsby
25.2 Thegradientofacurve 327
determinants 268
25.3 Differentiationfromfirstprinciples 328
21.3 Solutionofsimultaneousequations
25.4 Differentiationofcommonfunctions 329
usingCramer’srule 271
25.5 Differentiationofaproduct 332
21.4 Solutionofsimultaneousequations
usingtheGaussianeliminationmethod 272 25.6 Differentiationofaquotient 334
21.5 Stiffnessmatrix 274 25.7 Functionofafunction 335
21.6 Eigenvaluesandeigenvectors 275 25.8 Successivedifferentiation 337
26 Someapplicationsofdifferentiation 340
RevisionTest6 281 26.1 Ratesofchange 340
26.2 Velocityandacceleration 342
26.3 TheNewton–Raphsonmethod 345
26.4 Turningpoints 348
SectionF Vectorgeometry 283
26.5 Practicalproblemsinvolvingmaximum
andminimumvalues 351
26.6 Pointsofinflexion 355
22 Vectors 285 26.7 Tangentsandnormals 357
22.1 Introduction 285
26.8 Smallchanges 358
22.2 Scalarsandvectors 285
22.3 Drawingavector 286
RevisionTest8 362
22.4 Additionofvectorsbydrawing 287
