Bird's Engineering Mathematics

759 Pages·2021·50.49 MB·English

by  John Bird

Bird’s 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,test,andevaluateintegratedsystemsforman-
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
coordination.
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
Aerospace engineers require mathematics to perform mathematical tools such as differential equations, ten-
a variety of engineering work in designing, construct- soranalysis,fieldtheory,numericalmethodsandoper-
ing and testing aircraft, missiles and spacecraft; they ationsresearch.
conduct basic and applied research to evaluate adapt- Knowledgeofmathematicsisthereforeneededbyeach
abilityofmaterialsandequipmenttoaircraftdesignand oftheengineeringdisciplineslistedabove.
manufacture and recommend improvements in testing
Itisintendedthatthistext–Bird’sEngineeringMathe-
equipmentandtechniques.
matics–willprovideastepbystepapproachtolearning
Nuclear engineers require mathematics to conduct fundamentalmathematicsneededforyourengineering
research on nuclear engineering problems or apply studies.
Nowinitsninthedition,Bird’sEngineeringMathematicshashelpedthousandsofstudentstosucceedintheirexams.
Mathematicaltheoriesareexplainedinastraightforwardmanner,supportedbypracticalengineeringexamplesand
applicationstoensurethatreaderscanrelatetheorytopractice.Some1,300engineeringsituations/problemshave
been ‘flagged-up’ to help demonstrate that engineering cannot be fully understood without a good knowledge of
mathematics.
Theextensiveandthoroughtopiccoveragemakesthisagreattextforarangeoflevel2and3engineeringcourses–
suchasforaeronautical,construction,electrical,electronic,mechanical,manufacturingengineeringandvehicletech-
nology – including for BTEC First, National and Diploma syllabuses, City & Guilds Technician Certificate and
Diplomasyllabuses,andevenforGCSEandA-levelrevision.
Itscompanionwebsiteatwww.routledge.com/cw/birdprovidesresourcesforbothstudentsandlecturers,includ-
ingfullsolutionsforall2,000furtherquestions,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 the RoyalNavy’sDefenceCollegeofMarine Engineeringin the
DefenceCollegeofTechnicalTrainingatH.M.S.Sultan,Gosport,Hampshire,UK,oneofthelargestengineering
trainingestablishmentsinEurope.
In memory of Elizabeth

Bird’s Engineering Mathematics
Ninth Edition
John Bird
Nintheditionpublished2021
byRoutledge
2ParkSquare,MiltonPark,Abingdon,OxonOX144RN
andbyRoutledge
52VanderbiltAvenue,NewYork,NY10017
RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness
©2021JohnBird
TherightofJohnBirdtobeidentifiedastheauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78
oftheCopyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyformorbyanyelectronic,mechanical,
orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor
retrievalsystem,withoutpermissioninwritingfromthepublishers.
Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentification
andexplanationwithoutintenttoinfringe.
FirsteditionpublishedbyNewnes1999
EightheditionpublishedbyRoutledge2017
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
Acatalogrecordhasbeenrequestedforthisbook
ISBN:978-0-367-64379-9(hbk)
ISBN:978-0-367-64378-2(pbk)
ISBN:978-1-003-12423-8(ebk)
TypesetinTimes
byKnowledgeWorksGlobalLtd.
Visitthecompanionwebsite:www.routledge.com/cw/bird
Contents
Preface xiv 7 Partialfractions 73
7.1 Introductiontopartialfractions 73
Section1 Numberandalgebra 1 7.2 Partialfractionswithlinearfactors 74
7.3 Partialfractionswithrepeatedlinearfactors 76
7.4 Partialfractionswithquadraticfactors 77
1 Revisionoffractions,decimalsandpercentages 3
1.1 Fractions 3
1.2 Ratioandproportion 6 8 Solvingsimpleequations 80
1.3 Decimals 7 8.1 Expressions,equationsandidentities 80
1.4 Percentages 9 8.2 Workedproblemsonsimpleequations 81
8.3 Furtherworkedproblemsonsimple
equations 82
2 Indices,engineeringnotationandmetric
8.4 Practicalproblemsinvolvingsimple
conversions 13
equations 84
2.1 Indices 13
8.5 Furtherpracticalproblemsinvolving
2.2 Standardform 17
simpleequations 85
2.3 Engineeringnotationandcommonprefixes 18
2.4 Metricconversions 20
2.5 Metric-US/imperialconversions 23 RevisionTest2 88
3 Binary,octalandhexadecimalnumbers 30
9 Transpositionofformulae 89
3.1 Introduction 30
9.1 Introductiontotranspositionof
3.2 Binarynumbers 31
formulae 89
3.3 Octalnumbers 34
9.2 Workedproblemsontranspositionof
3.4 Hexadecimalnumbers 36
formulae 89
9.3 Furtherworkedproblemson
4 Calculationsandevaluationofformulae 41 transpositionofformulae 91
4.1 Errorsandapproximations 41 9.4 Harderworkedproblemson
4.2 Useofcalculator 43 transpositionofformulae 93
4.3 Conversiontablesandcharts 45
4.4 Evaluationofformulae 46
10 Solvingsimultaneousequations 98
10.1 Introductiontosimultaneous
RevisionTest1 52 equations 98
10.2 Workedproblemsonsimultaneous
equationsintwounknowns 98
5 Algebra 54 10.3 Furtherworkedproblemson
5.1 Basicoperations 54 simultaneousequations 100
5.2 Lawsofindices 56 10.4 Moredifficultworkedproblemson
5.3 Bracketsandfactorisation 58 simultaneousequations 102
5.4 Fundamentallawsandprecedence 60 10.5 Practicalproblemsinvolving
5.5 Directandinverseproportionality 62 simultaneousequations 104
6 Furtheralgebra 66 11 Solvingquadraticequations 109
6.1 Polynomialdivision 66 11.1 Introductiontoquadraticequations 109
6.2 Thefactortheorem 68 11.2 Solutionofquadraticequationsby
6.3 Theremaindertheorem 70 factorisation 110
viii Contents
11.3 Solutionofquadraticequationsby Section2 Trigonometry 167
‘completingthesquare’ 111
11.4 Solutionofquadraticequationsbyformula 113 17 Introductiontotrigonometry 169
11.5 Practicalproblemsinvolvingquadratic 17.1 Trigonometry 169
equations 114 17.2 ThetheoremofPythagoras 170
11.6 Thesolutionoflinearandquadratic 17.3 Trigonometricratiosofacuteangles 171
equationssimultaneously 117 17.4 Fractionalandsurdformsof
trigonometricratios 173
12 Inequalities 120
17.5 Evaluatingtrigonometricratiosofany
12.1 Introductiontoinequalities 120
angles 174
12.2 Simpleinequalities 121
17.6 Solutionofright-angledtriangles 178
12.3 Inequalitiesinvolvingamodulus 121
17.7 Angleofelevationanddepression 179
12.4 Inequalitiesinvolvingquotients 122
17.8 Trigonometricapproximationsforsmall
12.5 Inequalitiesinvolvingsquarefunctions 123 angles 181
12.6 Quadraticinequalities 124
18 Trigonometricwaveforms 184
13 Logarithms 126 18.1 Graphsoftrigonometricfunctions 184
13.1 Introductiontologarithms 126 18.2 Anglesofanymagnitude 185
13.2 Lawsoflogarithms 128 18.3 Theproductionofasineandcosinewave 187
13.3 Indicialequations 130 18.4 Sineandcosinecurves 188
13.4 Graphsoflogarithmicfunctions 132 18.5 SinusoidalformAsin(!t(cid:6)(cid:11)) 192
18.6 Waveformharmonics 194
RevisionTest3 134
19 Cartesianandpolarco-ordinates 196
19.1 Introduction 197
14 Exponentialfunctions 135 19.2 ChangingfromCartesianintopolar
14.1 Introductiontoexponentialfunctions 135 co-ordinates 197
14.2 Thepowerseriesforex 136 19.3 ChangingfrompolarintoCartesian
co-ordinates 198
14.3 Graphsofexponentialfunctions 138
19.4 UseofPol/Recfunctionsoncalculators 200
14.4 Napierianlogarithms 139
14.5 Lawsofgrowthanddecay 142
RevisionTest5 202
15 Numbersequences 147
15.1 Arithmeticprogressions 147
20 Trianglesandsomepracticalapplications 203
15.2 Workedproblemsonarithmetic
20.1 Sineandcosinerules 203
progressions 148
20.2 Areaofanytriangle 204
15.3 Furtherworkedproblemsonarithmetic
20.3 Workedproblemsonthesolutionof
progressions 149
trianglesandtheirareas 204
15.4 Geometricprogressions 150
20.4 Furtherworkedproblemsonthesolution
15.5 Workedproblemsongeometric
oftrianglesandtheirareas 206
progressions 151
20.5 Practicalsituationsinvolving
15.6 Furtherworkedproblemsongeometric
trigonometry 207
progressions 152
20.6 Furtherpracticalsituationsinvolving
15.7 Combinationsandpermutations 154
trigonometry 209
16 Thebinomialseries 156 21 Trigonometricidentitiesandequations 214
16.1 Pascal’striangle 156 21.1 Trigonometricidentities 214
16.2 Thebinomialseries 158 21.2 Workedproblemsontrigonometric
16.3 Workedproblemsonthebinomialseries 158 identities 215
16.4 Furtherworkedproblemsonthe 21.3 Trigonometricequations 216
binomialseries 159 21.4 Workedproblems(i)ontrigonometric
16.5 Practicalproblemsinvolvingthe equations 217
binomialtheorem 162 21.5 Workedproblems(ii)ontrigonometric
RevisionTest4 165 equations 218
Contents ix
21.6 Workedproblems(iii)ontrigonometric 26 Irregularareasandvolumesandmeanvalues
equations 219 ofwaveforms 276
21.7 Workedproblems(iv)ontrigonometric 26.1 Areaofirregularfigures 277
equations 219 26.2 Volumesofirregularsolids 279
26.3 Themeanoraveragevalue
22 Compoundangles 222
ofawaveform 280
22.1 Compoundangleformulae 222
22.2 Conversionofasin!t+bcos!tinto
RevisionTest7 285
Rsin(!t+(cid:11)) 224
22.3 Doubleangles 228
22.4 Changingproductsofsinesandcosines Section4 Graphs 287
intosumsordifferences 229
22.5 Changingsumsordifferencesofsines 27 Straightlinegraphs 289
andcosinesintoproducts 230 27.1 Introductiontographs 289
27.2 Thestraightlinegraph 290
RevisionTest6 233 27.3 Practicalproblemsinvolvingstraight
linegraphs 295
28 Reductionofnon-linearlawstolinearform 304
Section3 Areasandvolumes 235
28.1 Determinationoflaw 304
28.2 Determinationoflawinvolving
23 Areasofcommonshapes 237 logarithms 307
23.1 Introduction 237
29 Graphswithlogarithmicscales 313
23.2 Propertiesofquadrilaterals 238
29.1 Logarithmicscales 313
23.3 Areasofcommonshapes 238
29.2 Graphsoftheformy=axn 314
23.4 Workedproblemsonareasofcommon
29.3 Graphsoftheformy=abx 317
shapes 239
29.4 Graphsoftheformy=aekx 318
23.5 Furtherworkedproblemsonareasof
planefigures 242 30 Graphicalsolutionofequations 321
23.6 Workedproblemsonareasofcomposite 30.1 Graphicalsolutionofsimultaneous
figures 243 equations 321
23.7 Areasofsimilarshapes 245 30.2 Graphicalsolutionofquadraticequations 323
30.3 Graphicalsolutionoflinearand
24 Thecircleanditsproperties 247
quadraticequationssimultaneously 326
24.1 Introduction 247
30.4 Graphicalsolutionofcubicequations 327
24.2 Propertiesofcircles 247
24.3 Radiansanddegrees 249 31 Functionsandtheircurves 330
24.4 Arclengthandareaofcirclesand 31.1 Standardcurves 330
sectors 250 31.2 Simpletransformations 333
24.5 Workedproblemsonarclengthandarea 31.3 Periodicfunctions 337
ofcirclesandsectors 250 31.4 Continuousanddiscontinuousfunctions 337
24.6 Theequationofacircle 254 31.5 Evenandoddfunctions 338
25 Volumesandsurfaceareasofcommonsolids 257 31.6 Inversefunctions 339
25.1 Introduction 257
25.2 Volumesandsurfaceareasofregular RevisionTest8 343
solids 258
25.3 Workedproblemsonvolumesand
Section5 Complexnumbers 345
surfaceareasofregularsolids 258
25.4 Furtherworkedproblemsonvolumes
32 Complexnumbers 347
andsurfaceareasofregularsolids 260
32.1 Cartesiancomplexnumbers 347
25.5 Volumesandsurfaceareasoffrustaof
32.2 TheArganddiagram 349
pyramidsandcones 266
32.3 Additionandsubtractionofcomplex
25.6 Thefrustumandzoneofasphere 269
numbers 349
25.7 Prismoidalrule 272
32.4 Multiplicationanddivisionofcomplex
25.8 Volumesofsimilarshapes 274 numbers 350