Higher Engineering Mathematics, Sixth Edition

705 Pages·2010·5.77 MB·English

Higher Engineering Mathematics
In memory of Elizabeth
Higher Engineering Mathematics
Sixth Edition
John Bird
,BSc(Hons), CMath, CEng, CSci, FIMA,FIET, MIEE, FIIE,FCollT
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD
PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
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10 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1
Contents
Preface xiii 6 Arithmeticandgeometricprogressions 51
6.1 Arithmeticprogressions 51
Syllabusguidance xv
6.2 Workedproblemsonarithmetic
1 Algebra 1 progressions 51
1.1 Introduction 1 6.3 Furtherworkedproblemsonarithmetic
1.2 Revisionofbasiclaws 1 progressions 52
1.3 Revisionofequations 3 6.4 Geometricprogressions 54
1.4 Polynomialdivision 6 6.5 Workedproblemsongeometric
1.5 Thefactortheorem 8 progressions 55
1.6 Theremaindertheorem 10 6.6 Furtherworkedproblemsongeometric
progressions 56
2 Partialfractions 13 7 Thebinomialseries 58
2.1 Introductiontopartialfractions 13 7.1 Pascal’striangle 58
2.2 Workedproblemsonpartialfractionswith 7.2 Thebinomialseries 59
linearfactors 13
7.3 Workedproblemsonthebinomialseries 59
2.3 Workedproblemsonpartialfractionswith
7.4 Furtherworkedproblemsonthebinomial
repeatedlinearfactors 16 series 62
2.4 Workedproblemsonpartialfractionswith 7.5 Practicalproblemsinvolvingthebinomial
quadraticfactors 17
theorem 64
3 Logarithms 20 RevisionTest2 67
3.1 Introductiontologarithms 20
3.2 Lawsoflogarithms 22
8 Maclaurin’sseries 68
3.3 Indicialequations 24 8.1 Introduction 68
3.4 Graphsoflogarithmicfunctions 25 8.2 DerivationofMaclaurin’stheorem 68
8.3 ConditionsofMaclaurin’sseries 69
4 Exponentialfunctions 27 8.4 WorkedproblemsonMaclaurin’sseries 69
4.1 Introductiontoexponentialfunctions 27 8.5 NumericalintegrationusingMaclaurin’s
4.2 Thepowerseriesforex 28 series 73
4.3 Graphsofexponentialfunctions 29 8.6 Limitingvalues 74
4.4 Napierianlogarithms 31
9 Solvingequationsbyiterativemethods 77
4.5 Lawsofgrowthanddecay 34
9.1 Introductiontoiterativemethods 77
4.6 Reductionofexponentiallawsto
9.2 Thebisectionmethod 77
linearform 37
9.3 Analgebraicmethodofsuccessive
approximations 81
RevisionTest1 40
9.4 TheNewton-Raphsonmethod 84
10 Binary,octalandhexadecimal 87
5 Hyperbolicfunctions 41
10.1 Introduction 87
5.1 Introductiontohyperbolicfunctions 41
10.2 Binarynumbers 87
5.2 Graphsofhyperbolicfunctions 43
10.3 Octalnumbers 90
5.3 Hyperbolicidentities 45
10.4 Hexadecimalnumbers 92
5.4 Solvingequationsinvolvinghyperbolic
functions 47
RevisionTest3 96
5.5 Seriesexpansionsforcoshxandsinhx 49
vi
Contents
11 Introductiontotrigonometry 97 15.5 Workedproblems(ii)ontrigonometric
11.1 Trigonometry 97 equations 156
11.2 ThetheoremofPythagoras 97 15.6 Workedproblems(iii)ontrigonometric
11.3 Trigonometricratiosofacuteangles 98 equations 157
11.4 Evaluatingtrigonometricratios 100 15.7 Workedproblems(iv)ontrigonometric
11.5 Solutionofright-angledtriangles 105 equations 157
11.6 Anglesofelevationanddepression 106
11.7 Sineandcosinerules 108 16 Therelationshipbetweentrigonometricand
hyperbolicfunctions 159
11.8 Areaofanytriangle 108
16.1 Therelationshipbetweentrigonometric
11.9 Workedproblemsonthesolutionof
andhyperbolicfunctions 159
trianglesandfindingtheirareas 109
16.2 Hyperbolicidentities 160
11.10 Furtherworkedproblemsonsolving
trianglesandfindingtheirareas 110
17 Compoundangles 163
11.11 Practicalsituationsinvolving
17.1 Compoundangleformulae 163
trigonometry 111
17.2 Conversionofasinωt+bcosωt into
11.12 Furtherpracticalsituationsinvolving
Rsin(ωt+α) 165
trigonometry 113
17.3 Doubleangles 169
12 Cartesianandpolarco-ordinates 117 17.4 Changingproductsofsinesandcosines
12.1 Introduction 117 intosumsordifferences 170
12.2 ChangingfromCartesianintopolar 17.5 Changingsumsordifferencesofsinesand
co-ordinates 117 cosinesintoproducts 171
12.3 ChangingfrompolarintoCartesian 17.6 Powerwaveformsina.c.circuits 173
co-ordinates 119
12.4 UseofPol/Recfunctionsoncalculators 120 RevisionTest5 177
13 Thecircleanditsproperties 122
13.1 Introduction 122 18 Functionsandtheircurves 178
13.2 Propertiesofcircles 122 18.1 Standardcurves 178
13.3 Radiansanddegrees 123 18.2 Simpletransformations 181
13.4 Arclengthandareaofcirclesandsectors 124 18.3 Periodicfunctions 186
13.5 Theequationofacircle 127 18.4 Continuousanddiscontinuousfunctions 186
13.6 Linearandangularvelocity 129 18.5 Evenandoddfunctions 186
13.7 Centripetalforce 130 18.6 Inversefunctions 188
18.7 Asymptotes 190
18.8 Briefguidetocurvesketching 196
RevisionTest4 133
18.9 Workedproblemsoncurvesketching 197
14 Trigonometricwaveforms 134 19 Irregularareas,volumesandmeanvaluesof
14.1 Graphsoftrigonometricfunctions 134 waveforms 203
14.2 Anglesofanymagnitude 135 19.1 Areasofirregularfigures 203
14.3 Theproductionofasineandcosinewave 137 19.2 Volumesofirregularsolids 205
14.4 Sineandcosinecurves 138 19.3 Themeanoraveragevalueofawaveform 206
14.5 Sinusoidalform Asin(ωt±α) 143
14.6 Harmonicsynthesiswithcomplex
RevisionTest6 212
waveforms 146
15 Trigonometricidentitiesandequations 152 20 Complexnumbers 213
15.1 Trigonometricidentities 152 20.1 Cartesiancomplexnumbers 213
15.2 Workedproblemsontrigonometric 20.2 TheArganddiagram 214
identities 152 20.3 Additionandsubtractionofcomplex
15.3 Trigonometricequations 154 numbers 214
15.4 Workedproblems(i)ontrigonometric 20.4 Multiplicationanddivisionofcomplex
equations 154 numbers 216
vii
Contents
20.5 Complexequations 217 25.4 Determiningresultantphasorsbythesine
20.6 Thepolarformofacomplexnumber 218 andcosinerules 268
20.7 Multiplicationanddivisioninpolarform 220 25.5 Determiningresultantphasorsby
20.8 Applicationsofcomplexnumbers 221 horizontalandverticalcomponents 270
25.6 Determiningresultantphasorsbycomplex
numbers 272
21 DeMoivre’stheorem 225
21.1 Introduction 225
26 Scalarandvectorproducts 275
21.2 Powersofcomplexnumbers 225
26.1 Theunittriad 275
21.3 Rootsofcomplexnumbers 226
26.2 Thescalarproductoftwovectors 276
21.4 Theexponentialformofacomplex
26.3 Vectorproducts 280
number 228
26.4 Vectorequationofaline 283
22 Thetheoryofmatricesanddeterminants 231
RevisionTest8 286
22.1 Matrixnotation 231
22.2 Addition,subtractionandmultiplication
ofmatrices 231
27 Methodsofdifferentiation 287
22.3 Theunitmatrix 235 27.1 Introductiontocalculus 287
22.4 Thedeterminantofa2by2matrix 235 27.2 Thegradientofacurve 287
22.5 Theinverseorreciprocalofa2by2matrix 236 27.3 Differentiationfromfirstprinciples 288
22.6 Thedeterminantofa3by3matrix 237 27.4 Differentiationofcommonfunctions 289
22.7 Theinverseorreciprocalofa3by3matrix 239 27.5 Differentiationofaproduct 292
27.6 Differentiationofaquotient 293
23 Thesolutionofsimultaneousequationsby 27.7 Functionofafunction 295
matricesanddeterminants 241 27.8 Successivedifferentiation 296
23.1 Solutionofsimultaneousequationsby
matrices 241 28 Someapplicationsofdifferentiation 299
23.2 Solutionofsimultaneousequationsby 28.1 Ratesofchange 299
determinants 243 28.2 Velocityandacceleration 300
23.3 Solutionofsimultaneousequationsusing 28.3 Turningpoints 303
Cramersrule 247
28.4 Practicalproblemsinvolvingmaximum
23.4 Solutionofsimultaneousequationsusing andminimumvalues 307
theGaussianeliminationmethod 248
28.5 Tangentsandnormals 311
28.6 Smallchanges 312
RevisionTest7 250
29 Differentiationofparametricequations 315
29.1 Introductiontoparametricequations 315
24 Vectors 251
29.2 Somecommonparametricequations 315
24.1 Introduction 251
29.3 Differentiationinparameters 315
24.2 Scalarsandvectors 251
29.4 Furtherworkedproblemson
24.3 Drawingavector 251
differentiationofparametricequations 318
24.4 Additionofvectorsbydrawing 252
24.5 Resolvingvectorsintohorizontaland
30 Differentiationofimplicitfunctions 320
verticalcomponents 254
30.1 Implicitfunctions 320
24.6 Additionofvectorsbycalculation 255
30.2 Differentiatingimplicitfunctions 320
24.7 Vectorsubtraction 260
30.3 Differentiatingimplicitfunctions
24.8 Relativevelocity 262 containingproductsandquotients 321
24.9 i,j andknotation 263 30.4 Furtherimplicitdifferentiation 322
25 Methodsofaddingalternatingwaveforms 265 31 Logarithmicdifferentiation 325
25.1 Combinationoftwoperiodicfunctions 265 31.1 Introductiontologarithmicdifferentiation 325
25.2 Plottingperiodicfunctions 265 31.2 Lawsoflogarithms 325
25.3 Determiningresultantphasorsbydrawing 267 31.3 Differentiationoflogarithmicfunctions 325
viii
Contents
31.4 Differentiationoffurtherlogarithmic 38 Someapplicationsofintegration 375
functions 326 38.1 Introduction 375
31.5 Differentiationof[f(x)]x 328 38.2 Areasunderandbetweencurves 375
38.3 Meanandr.m.s.values 377
RevisionTest9 330 38.4 Volumesofsolidsofrevolution 378
38.5 Centroids 380
38.6 TheoremofPappus 381
32 Differentiationofhyperbolicfunctions 331
32.1 Standarddifferentialcoefficientsof 38.7 Secondmomentsofareaofregular
hyperbolicfunctions 331 sections 383
32.2 Furtherworkedproblemson
39 Integrationusingalgebraicsubstitutions 392
differentiationofhyperbolicfunctions 332
39.1 Introduction 392
33 Differentiationofinversetrigonometricand 39.2 Algebraicsubstitutions 392
hyperbolicfunctions 334 39.3 Workedproblemsonintegrationusing
33.1 Inversefunctions 334 algebraicsubstitutions 392
33.2 Differentiationofinversetrigonometric 39.4 Furtherworkedproblemsonintegration
functions 334 usingalgebraicsubstitutions 394
33.3 Logarithmicformsoftheinverse 39.5 Changeoflimits 395
hyperbolicfunctions 339
33.4 Differentiationofinversehyperbolic RevisionTest11 397
functions 341
34 Partialdifferentiation 345 40 Integrationusingtrigonometricandhyperbolic
34.1 Introductiontopartialderivatives 345 substitutions 398
34.2 Firstorderpartialderivatives 345 40.1 Introduction 398
34.3 Secondorderpartialderivatives 348 40.2 Workedproblemsonintegrationofsin2x,
cos2x,tan2x andcot2x 398
35 Totaldifferential,ratesofchangeandsmall
40.3 Workedproblemsonpowersofsinesand
changes 351
cosines 400
35.1 Totaldifferential 351
40.4 Workedproblemsonintegrationof
35.2 Ratesofchange 352
productsofsinesandcosines 401
35.3 Smallchanges 354
40.5 Workedproblemsonintegrationusingthe
sinθ substitution 402
36 Maxima,minimaandsaddlepointsforfunctions
40.6 Workedproblemsonintegrationusing
oftwovariables 357
tanθ substitution 404
36.1 Functionsoftwoindependentvariables 357
40.7 Workedproblemsonintegrationusingthe
36.2 Maxima,minimaandsaddlepoints 358
sinhθ substitution 404
36.3 Proceduretodeterminemaxima,minima
40.8 Workedproblemsonintegrationusingthe
andsaddlepointsforfunctionsoftwo
coshθ substitution 406
variables 359
36.4 Workedproblemsonmaxima,minima
41 Integrationusingpartialfractions 409
andsaddlepointsforfunctionsoftwo
41.1 Introduction 409
variables 359
41.2 Workedproblemsonintegrationusing
36.5 Furtherworkedproblemsonmaxima,
partialfractionswithlinearfactors 409
minimaandsaddlepointsforfunctionsof
41.3 Workedproblemsonintegrationusing
twovariables 361
partialfractionswithrepeatedlinear
factors 411
RevisionTest10 367
41.4 Workedproblemsonintegrationusing
partialfractionswithquadraticfactors 412
37 Standardintegration 368
37.1 Theprocessofintegration 368 θ
42 Thet=tan substitution 414
37.2 Thegeneralsolutionofintegralsofthe 2
formaxn 368 42.1 Introduction θ 414
37.3 Standardintegrals 369 42.2 Workedproblemsonthet=tan
2
37.4 Definiteintegrals 372 substitution 415
ix
Contents
θ
42.3 Furtherworkedproblemsonthet= tan 48 Linearfirstorderdifferentialequations 456
2 48.1 Introduction 456
substitution 416
48.2 Proceduretosolvedifferentialequations
dy
RevisionTest12 419 oftheform +Py=Q 457
dx
48.3 Workedproblemsonlinearfirstorder
differentialequations 457
43 Integrationbyparts 420 48.4 Furtherworkedproblemsonlinearfirst
43.1 Introduction 420 orderdifferentialequations 458
43.2 Workedproblemsonintegrationbyparts 420
43.3 Furtherworkedproblemsonintegration 49 Numericalmethodsforfirstorderdifferential
byparts 422 equations 461
49.1 Introduction 461
49.2 Euler’smethod 461
44 Reductionformulae 426
44.1 Introduction 426 49.3 WorkedproblemsonEuler’smethod 462
44.2 Usingred(cid:2)uctionformulaeforintegralsof 49.4 AnimprovedEulermethod 466
theform xnexdx 426 49.5 TheRunge-Kuttamethod 471
44.3 Usingred(cid:2)uctionformulae(cid:2)forintegralsof
theform xncosxdxand xn sinxdx 427 RevisionTest14 476
44.4 Usingred(cid:2)uctionformulae(cid:2)forintegralsof
theform sinnxdxand cosnxdx 429
50 Secondorderdifferentialequationsoftheform
44.5 Furtherreductionformulae 432
d2y dy
a +b +cy=0 477
dx2 dx
45 Numericalintegration 435
50.1 Introduction 477
45.1 Introduction 435
50.2 Proceduretosolvedifferentialequations
45.2 Thetrapezoidalrule 435
d2y dy
45.3 Themid-ordinaterule 437 oftheforma +b +cy=0 478
dx2 dx
45.4 Simpson’srule 439
50.3 Workedproblemsondifferentialequations
d2y dy
oftheforma +b +cy=0 478
RevisionTest13 443 dx2 dx
50.4 Furtherworkedproblemsonpractical
differentialequationsoftheform
d2y dy
46 Solutionoffirstorderdifferentialequationsby a +b +cy=0 480
separationofvariables 444 dx2 dx
46.1 Familyofcurves 444
51 Secondorderdifferentialequationsoftheform
46.2 Differentialequations 445
d2y dy
46.3 Thesolutionofequationsoftheform a +b +cy=f(x) 483
dy dx2 dx
= f(x) 445
dx 51.1 Complementaryfunctionandparticular
46.4 Thesolutionofequationsoftheform integral 483
dy
= f(y) 447 51.2 Proceduretosolvedifferentialequations
dx d2y dy
46.5 Thesolutionofequationsoftheform oftheforma +b +cy=f(x) 483
dy dx2 dx
= f(x)· f(y) 449 51.3 Workedproblemsondifferentialequations
dx d2y dy
oftheforma +b +cy=f(x)
dx2 dx
47 Homogeneousfirstorderdifferentialequations 452 where f(x)isaconstantorpolynomial 484
47.1 Introduction 452 51.4 Workedproblemsondifferentialequations
47.2 Proceduretosolvedifferentialequations d2y dy
dy oftheforma +b +cy=f(x)
oftheform P =Q 452 dx2 dx
dx where f(x)isanexponentialfunction 486
47.3 Workedproblemsonhomogeneousfirst
51.5 Workedproblemsondifferentialequations
orderdifferentialequations 452 d2y dy
47.4 Furtherworkedproblemsonhomogeneous oftheformadx2+bdx+cy=f(x)
firstorderdifferentialequations 454 where f(x)isasineorcosinefunction 488