Table Of Content

Higher Engineering Mathematics Whyisknowledgeofmathematicsimportantinengineering? A career in any engineering or scientific field will ciples and theory of nuclear science to problems con- requirebothbasicandadvancedmathematics.Without cernedwith release, control,and utilisation of nuclear mathematicsto determineprinciples,calculate dimen- energyandnuclearwastedisposal. sionsandlimits,explorevariations,proveconcepts,and Petroleum engineers require mathematics to devise soon,therewouldbenomobiletelephones,televisions, methods to improve oil and gas well production and stereosystems,videogames,microwaveovens,comput- determine the need for new or modified tool designs; ers,orvirtuallyanythingelectronic.Therewouldbeno they oversee drilling and offer technical advice to bridges,tunnels,roads,skyscrapers,automobiles,ships, achieveeconomicalandsatisfactoryprogress. planes,rocketsormostthingsmechanical.Therewould be no metals beyond the common ones, such as iron Industrial engineers require mathematics to design, and copper, no plastics, no synthetics. In fact, society develop,test,andevaluateintegratedsystemsforman- wouldmostcertainlybelessadvancedwithouttheuse agingindustrialproductionprocesses,includinghuman of mathematics throughout the centuries and into the work factors, quality control, inventorycontrol, logis- future. tics and material flow, cost analysis, and production co-ordination. Electrical engineers require mathematics to design, develop,test,orsupervisethemanufacturingandinstal- Environmental engineers require mathematics to lationofelectricalequipment,components,orsystems design, plan, or perform engineering duties in the forcommercial,industrial,military,orscientificuse. prevention, control, and remediation of environmental health hazards, using various engineering disci- Mechanicalengineersrequiremathematicstoperform plines; their work may include waste treatment, site engineering duties in planning and designing tools, remediation,orpollutioncontroltechnology. engines,machines,andothermechanicallyfunctioning equipment;theyoverseeinstallation,operation,mainte- Civil engineers require mathematics in all levels in nance,andrepairofsuchequipmentascentralisedheat, civil engineering – structural engineering, hydraulics gas,water,andsteamsystems. andgeotechnicalengineeringareallfieldsthatemploy mathematicaltoolssuchasdifferentialequations,tensor Aerospace engineers require mathematics to perform analysis,fieldtheory,numericalmethodsandoperations a variety of engineeringwork in designing,construct- research. ing, and testing aircraft, missiles, and spacecraft;they conduct basic and applied research to evaluate adapt- Knowledgeofmathematicsisthereforeneededbyeach abilityofmaterialsandequipmenttoaircraftdesignand oftheengineeringdisciplineslistedabove. manufacture and recommendimprovementsin testing Itisintendedthatthistext–HigherEngineeringMathe- equipmentandtechniques. matics–willprovideastep-by-stepapproachtolearning Nuclear engineers require mathematics to conduct fundamentalmathematicsneededforyourengineering researchonnuclearengineeringproblemsorapplyprin- studies. To Sue Higher Engineering Mathematics Seventh Edition John Bird ,BSc(Hons), CMath, CEng, CSci, FITE, FIMA, FCollT Routledge Taylor & Francis Group LONDON AND NEW YORK Seventheditionpublished2014 byRoutledge 2ParkSquare,MiltonPark,Abingdon,OxonOX144RN andbyRoutledge 711ThirdAvenue,NewYork,NY10017 RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness ©2014JohnBird TherightofJohnBirdtobeidentifiedasauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78 oftheCopyright,DesignsandPatentsAct1988. Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyformorbyanyelectronic,mechanical, orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor retrievalsystem,withoutpermissioninwritingfromthepublishers. Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentification andexplanationwithoutintenttoinfringe. FirsteditionpublishedbyElsevier1993 SixtheditionpublishedbyNewnes2010 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationData Bird,J.O. Higherengineeringmathematics/JohnBird,BSc(Hons),CMath,CEng,CSci,FITE,FIMA,FCoIIT.–Seventhedition. pagescm Includesindex. 1.Engineeringmathematics.I.Title. TA330.B522014 620.001’51–dc23 2013027617 ISBN:978-0-415-66282-6(pbk) ISBN:978-0-315-85882-1(ebk) TypesetinTimesby ServisFilmsettingLtd,Stockport,Cheshire Contents RevisionTest1 49 Preface xiii 6 Arithmeticandgeometricprogressions 50 Syllabusguidance xv 6.1 Arithmeticprogressions 50 6.2 Workedproblemsonarithmetic SectionA Numberandalgebra 1 progressions 51 6.3 Furtherworkedproblemsonarithmetic progressions 52 1 Algebra 3 6.4 Geometricprogressions 53 1.1 Introduction 3 6.5 Workedproblemsongeometric 1.2 Revisionofbasiclaws 3 progressions 54 1.3 Revisionofequations 5 6.6 Furtherworkedproblemsongeometric 1.4 Polynomialdivision 8 progressions 55 1.5 Thefactortheorem 10 1.6 Theremaindertheorem 12 7 Thebinomialseries 58 7.1 Pascal’striangle 58 2 Partialfractions 15 7.2 Thebinomialseries 60 2.1 Introductiontopartialfractions 15 7.3 Workedproblemsonthebinomialseries 60 2.2 Workedproblemsonpartialfractionswith linearfactors 16 7.4 Furtherworkedproblemsonthebinomial series 62 2.3 Workedproblemsonpartialfractionswith repeatedlinearfactors 18 7.5 Practicalproblemsinvolvingthebinomial theorem 64 2.4 Workedproblemsonpartialfractionswith quadraticfactors 20 8 Maclaurin’sseries 68 3 Logarithms 22 8.1 Introduction 69 3.1 Introductiontologarithms 22 8.2 DerivationofMaclaurin’stheorem 69 3.2 Lawsoflogarithms 24 8.3 ConditionsofMaclaurin’sseries 70 3.3 Indicialequations 27 8.4 WorkedproblemsonMaclaurin’sseries 70 3.4 Graphsoflogarithmicfunctions 28 8.5 NumericalintegrationusingMaclaurin’s series 73 4 Exponentialfunctions 29 8.6 Limitingvalues 75 4.1 Introductiontoexponentialfunctions 29 4.2 Thepowerseriesforex 30 RevisionTest2 78 4.3 Graphsofexponentialfunctions 32 4.4 Napierianlogarithms 33 9 Solvingequationsbyiterativemethods 79 4.5 Lawsofgrowthanddecay 36 9.1 Introductiontoiterativemethods 79 4.6 Reductionofexponentiallawsto 9.2 Thebisectionmethod 80 linearform 40 9.3 Analgebraicmethodofsuccessive approximations 83 5 Inequalities 43 9.4 TheNewton–Raphsonmethod 86 5.1 Introductiontoinequalities 43 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 93 5.6 Quadraticinequalities 47 10.4 Hexadecimalnumbers 95 vi Contents 11 Booleanalgebraandlogiccircuits 99 15 Trigonometricwaveforms 160 11.1 Booleanalgebraandswitchingcircuits 100 15.1 Graphsoftrigonometricfunctions 160 11.2 SimplifyingBooleanexpressions 104 15.2 Anglesofanymagnitude 161 11.3 LawsandrulesofBooleanalgebra 104 15.3 Theproductionofasineandcosinewave 164 11.4 DeMorgan’slaws 106 15.4 Sineandcosinecurves 165 11.5 Karnaughmaps 107 15.5 Sinusoidalform Asin(ωt±α) 169 11.6 Logiccircuits 111 15.6 Harmonicsynthesiswithcomplex 11.7 Universallogicgates 115 waveforms 172 RevisionTest3 118 16 Hyperbolicfunctions 178 16.1 Introductiontohyperbolicfunctions 178 16.2 Graphsofhyperbolicfunctions 180 SectionB Geometryandtrigonometry 119 16.3 Hyperbolicidentities 182 16.4 Solvingequationsinvolvinghyperbolic functions 184 12 Introductiontotrigonometry 121 16.5 Seriesexpansionsforcoshx andsinhx 186 12.1 Trigonometry 122 12.2 ThetheoremofPythagoras 122 12.3 Trigonometricratiosofacuteangles 123 17 Trigonometricidentitiesandequations 188 12.4 Evaluatingtrigonometricratios 125 17.1 Trigonometricidentities 188 12.5 Solutionofright-angledtriangles 129 17.2 Workedproblemsontrigonometric 12.6 Anglesofelevationanddepression 131 identities 189 12.7 Sineandcosinerules 132 17.3 Trigonometricequations 190 12.8 Areaofanytriangle 133 17.4 Workedproblems(i)ontrigonometric 12.9 Workedproblemsonthesolutionof equations 191 trianglesandfindingtheirareas 133 17.5 Workedproblems(ii)ontrigonometric 12.10 Furtherworkedproblemsonsolving equations 192 trianglesandfindingtheirareas 134 17.6 Workedproblems(iii)ontrigonometric 12.11 Practicalsituationsinvolving equations 193 trigonometry 136 17.7 Workedproblems(iv)ontrigonometric 12.12 Furtherpracticalsituationsinvolving equations 193 trigonometry 138 18 Therelationshipbetweentrigonometricand 13 Cartesianandpolarco-ordinates 141 hyperbolicfunctions 196 13.1 Introduction 142 18.1 Therelationshipbetweentrigonometric 13.2 ChangingfromCartesianintopolar andhyperbolicfunctions 196 co-ordinates 142 18.2 Hyperbolicidentities 197 13.3 ChangingfrompolarintoCartesian co-ordinates 144 13.4 UseofPol/Recfunctionsoncalculators 145 19 Compoundangles 200 19.1 Compoundangleformulae 200 19.2 Conversionofasinωt+bcosωt into 14 Thecircleanditsproperties 147 Rsin(ωt+α) 202 14.1 Introduction 147 19.3 Doubleangles 206 14.2 Propertiesofcircles 147 19.4 Changingproductsofsinesandcosines 14.3 Radiansanddegrees 149 intosumsordifferences 208 14.4 Arclengthandareaofcirclesandsectors 150 19.5 Changingsumsordifferencesofsinesand 14.5 Theequationofacircle 153 cosinesintoproducts 209 14.6 Linearandangularvelocity 154 19.6 Powerwaveformsina.c.circuits 210 14.7 Centripetalforce 156 RevisionTest4 158 RevisionTest5 214 Contents vii 24.5 Theinverseorreciprocalofa2by2matrix 280 SectionC Graphs 215 24.6 Thedeterminantofa3by3matrix 281 24.7 Theinverseorreciprocalofa3by3matrix 283 20 Functionsandtheircurves 217 20.1 Standardcurves 217 25 Applicationsofmatricesanddeterminants 285 20.2 Simpletransformations 220 25.1 Solutionofsimultaneousequationsby 20.3 Periodicfunctions 225 matrices 286 20.4 Continuousanddiscontinuousfunctions 225 25.2 Solutionofsimultaneousequationsby 20.5 Evenandoddfunctions 226 determinants 288 20.6 Inversefunctions 227 25.3 Solutionofsimultaneousequationsusing 20.7 Asymptotes 229 Cramer’srule 291 20.8 Briefguidetocurvesketching 235 25.4 Solutionofsimultaneousequationsusing 20.9 Workedproblemsoncurvesketching 236 theGaussianeliminationmethod 292 25.5 Eigenvaluesandeigenvectors 294 21 Irregularareas,volumesandmeanvaluesof waveforms 239 21.1 Areasofirregularfigures 239 RevisionTest7 300 21.2 Volumesofirregularsolids 242 21.3 Themeanoraveragevalueofawaveform 243 SectionF Vectorgeometry 301 RevisionTest6 248 26 Vectors 303 26.1 Introduction 303 SectionD Complexnumbers 249 26.2 Scalarsandvectors 303 26.3 Drawingavector 304 22 Complexnumbers 251 26.4 Additionofvectorsbydrawing 305 22.1 Cartesiancomplexnumbers 252 26.5 Resolvingvectorsintohorizontaland 22.2 TheArganddiagram 253 verticalcomponents 307 22.3 Additionandsubtractionofcomplex 26.6 Additionofvectorsbycalculation 308 numbers 253 26.7 Vectorsubtraction 312 22.4 Multiplicationanddivisionofcomplex 26.8 Relativevelocity 314 numbers 254 26.9 i,j andknotation 315 22.5 Complexequations 256 22.6 Thepolarformofacomplexnumber 257 22.7 Multiplicationanddivisioninpolarform 259 27 Methodsofaddingalternatingwaveforms 317 22.8 Applicationsofcomplexnumbers 260 27.1 Combinationoftwoperiodicfunctions 317 27.2 Plottingperiodicfunctions 318 23 DeMoivre’stheorem 264 27.3 Determiningresultantphasorsbydrawing 319 23.1 Introduction 265 27.4 Determiningresultantphasorsbythesine 23.2 Powersofcomplexnumbers 265 andcosinerules 321 23.3 Rootsofcomplexnumbers 266 27.5 Determiningresultantphasorsby 23.4 Theexponentialformofacomplex horizontalandverticalcomponents 322 number 268 27.6 Determiningresultantphasorsbycomplex 23.5 Introductiontolocusproblems 269 numbers 324 SectionE Matricesanddeterminants 273 28 Scalarandvectorproducts 328 28.1 Theunittriad 328 24 Thetheoryofmatricesanddeterminants 275 28.2 Thescalarproductoftwovectors 329 24.1 Matrixnotation 275 28.3 Vectorproducts 333 24.2 Addition,subtractionandmultiplication 28.4 Vectorequationofaline 337 ofmatrices 276 24.3 Theunitmatrix 279 RevisionTest8 339 24.4 Thedeterminantofa2by2matrix 279 viii Contents 35.2 Differentiationofinversetrigonometric SectionG Differentialcalculus 341 functions 397 35.3 Logarithmicformsoftheinverse 29 Methodsofdifferentiation 343 hyperbolicfunctions 400 29.1 Introductiontocalculus 343 35.4 Differentiationofinversehyperbolic 29.2 Thegradientofacurve 343 functions 402 29.3 Differentiationfromfirstprinciples 344 29.4 Differentiationofcommonfunctions 345 36 Partialdifferentiation 406 29.5 Differentiationofaproduct 348 36.1 Introductiontopartialderivatives 406 29.6 Differentiationofaquotient 350 36.2 First-orderpartialderivatives 406 29.7 Functionofafunction 351 36.3 Second-orderpartialderivatives 409 29.8 Successivedifferentiation 353 37 Totaldifferential,ratesofchangeandsmall 30 Someapplicationsofdifferentiation 355 changes 412 30.1 Ratesofchange 355 37.1 Totaldifferential 412 30.2 Velocityandacceleration 357 37.2 Ratesofchange 413 30.3 Turningpoints 360 37.3 Smallchanges 416 30.4 Practicalproblemsinvolvingmaximum 38 Maxima,minimaandsaddlepointsforfunctions andminimumvalues 363 oftwovariables 419 30.5 Pointsofinflexion 367 38.1 Functionsoftwoindependentvariables 419 30.6 Tangentsandnormals 369 38.2 Maxima,minimaandsaddlepoints 420 30.7 Smallchanges 370 38.3 Proceduretodeterminemaxima,minima 31 Differentiationofparametricequations 373 andsaddlepointsforfunctionsoftwo 31.1 Introductiontoparametricequations 373 variables 421 31.2 Somecommonparametricequations 374 38.4 Workedproblemsonmaxima,minima 31.3 Differentiationinparameters 374 andsaddlepointsforfunctionsoftwo variables 421 31.4 Furtherworkedproblemson differentiationofparametricequations 376 38.5 Furtherworkedproblemsonmaxima, minimaandsaddlepointsforfunctionsof 32 Differentiationofimplicitfunctions 379 twovariables 424 32.1 Implicitfunctions 379 32.2 Differentiatingimplicitfunctions 379 RevisionTest10 429 32.3 Differentiatingimplicitfunctions containingproductsandquotients 380 32.4 Furtherimplicitdifferentiation 381 SectionH Integralcalculus 431 33 Logarithmicdifferentiation 385 39 Standardintegration 433 33.1 Introductiontologarithmicdifferentiation 385 39.1 Theprocessofintegration 433 33.2 Lawsoflogarithms 385 39.2 Thegeneralsolutionofintegralsofthe 33.3 Differentiationoflogarithmicfunctions 386 formaxn 434 33.4 Differentiationoffurtherlogarithmic 39.3 Standardintegrals 434 functions 386 39.4 Definiteintegrals 437 33.5 Differentiationof[f(x)]x 388 40 Someapplicationsofintegration 440 40.1 Introduction 441 RevisionTest9 391 40.2 Areasunderandbetweencurves 441 40.3 Meanandrmsvalues 442 34 Differentiationofhyperbolicfunctions 392 40.4 Volumesofsolidsofrevolution 443 34.1 Standarddifferentialcoefficientsof hyperbolicfunctions 392 40.5 Centroids 445 34.2 Furtherworkedproblemson 40.6 TheoremofPappus 447 differentiationofhyperbolicfunctions 393 40.7 Secondmomentsofareaofregular sections 449 35 Differentiationofinversetrigonometricand hyperbolicfunctions 395 41 Integrationusingalgebraicsubstitutions 457 35.1 Inversefunctions 395 41.1 Introduction 457 Contents ix 41.2 Algebraicsubstitutions 457 46.2 Usingred(cid:2)uctionformulaeforintegralsof 41.3 Workedproblemsonintegrationusing theform xnexdx 491 algebraicsubstitutions 458 46.3 Usingred(cid:2)uctionformulae(cid:2)forintegralsof 41.4 Furtherworkedproblemsonintegration theform xncosxdx and xn sinxdx 492 usingalgebraicsubstitutions 459 46.4 Usingred(cid:2)uctionformulae(cid:2)forintegralsof 41.5 Changeoflimits 460 theform sinnxdx and cosnxdx 495 46.5 Furtherreductionformulae 497 RevisionTest11 462 47 Doubleandtripleintegrals 500 47.1 Doubleintegrals 500 42 Integrationusingtrigonometricandhyperbolic 47.2 Tripleintegrals 502 substitutions 463 42.1 Introduction 463 48 Numericalintegration 505 42.2 Workedproblemsonintegrationofsin2x, 48.1 Introduction 505 cos2x,tan2x andcot2x 463 48.2 Thetrapezoidalrule 505 42.3 Workedproblemsonintegrationofpowers 48.3 Themid-ordinaterule 508 ofsinesandcosines 466 48.4 Simpson’srule 509 42.4 Workedproblemsonintegrationof productsofsinesandcosines 467 RevisionTest13 514 42.5 Workedproblemsonintegrationusingthe sinθ substitution 468 42.6 Workedproblemsonintegrationusing SectionI Differentialequations 515 tanθ substitution 469 42.7 Workedproblemsonintegrationusingthe sinhθ substitution 470 49 Solutionoffirst-orderdifferentialequationsby separationofvariables 517 42.8 Workedproblemsonintegrationusingthe coshθ substitution 472 49.1 Familyofcurves 517 49.2 Differentialequations 518 43 Integrationusingpartialfractions 474 49.3 Thesolutionofequationsoftheform 43.1 Introduction 474 dy = f(x) 519 43.2 Workedproblemsonintegrationusing dx partialfractionswithlinearfactors 474 49.4 Thesolutionofequationsoftheform dy 43.3 Workedproblemsonintegrationusing = f(y) 520 partialfractionswithrepeatedlinear dx 49.5 Thesolutionofequationsoftheform factors 476 dy 43.4 Workedproblemsonintegrationusing = f(x)· f(y) 522 dx partialfractionswithquadraticfactors 477 50 Homogeneousfirst-orderdifferentialequations 526 θ 44 Thet=tan substitution 479 50.1 Introduction 526 2 50.2 Proceduretosolvedifferentialequations 44.1 Introduction 479 θ dy 44.2 Workedproblemsonthet=tan oftheform P =Q 526 2 dx substitution 480 50.3 Workedproblemsonhomogeneous θ 44.3 Furtherworkedproblemsonthet= tan first-orderdifferentialequations 527 2 50.4 Furtherworkedproblemsonhomogeneous substitution 481 first-orderdifferentialequations 528 RevisionTest12 484 51 Linearfirst-orderdifferentialequations 530 51.1 Introduction 530 45 Integrationbyparts 485 51.2 Proceduretosolvedifferentialequations 45.1 Introduction 485 dy oftheform +Py=Q 531 45.2 Workedproblemsonintegrationbyparts 485 dx 51.3 Workedproblemsonlinearfirst-order 45.3 Furtherworkedproblemsonintegration differentialequations 531 byparts 487 51.4 Furtherworkedproblemsonlinear 46 Reductionformulae 491 first-orderdifferentialequations 532 46.1 Introduction 491