Higher Engineering Mathematics

Preface xiii
Syllabus guidance xv
1 Algebra 1
1.1 Introduction 1
1.2 Revision of basic laws 1
1.3 Revision of equations 3
1.4 Polynomial division 6
1.5 The factor theorem 8
1.6 The remainder theorem 10
2 Partial fractions 13
2.1 Introduction to partial fractions 13
2.2 Worked problems on partial fractions with
linear factors 13
2.3 Worked problems on partial fractions with
repeated linear factors 16
2.4 Worked problems on partial fractions with
quadratic factors 17
3 Logarithms 20
3.1 Introduction to logarithms 20
3.2 Laws of logarithms 22
3.3 Indicial equations 24
3.4 Graphs of logarithmic functions 25
4 Exponential functions 27
4.1 Introduction to exponential functions 27
4.2 The power series for ex 28
4.3 Graphs of exponential functions 29
4.4 Napierian logarithms 31
4.5 Laws of growth and decay 34
4.6 Reduction of exponential laws to
linear form 37
Revision Test 1 40
5 Hyperbolic functions 41
5.1 Introduction to hyperbolic functions 41
5.2 Graphs of hyperbolic functions 43
5.3 Hyperbolic identities 45
5.4 Solving equations involving hyperbolic
functions 47
5.5 Series expansions for cosh x and sinh x 49
6 Arithmetic and geometric progressions 51
6.1 Arithmetic progressions 51
6.2 Worked problems on arithmetic
progressions 51
6.3 Further worked problems on arithmetic
progressions 52
6.4 Geometric progressions 54
6.5 Worked problems on geometric
progressions 55
6.6 Further worked problems on geometric
progressions 56
7 The binomial series 58
7.1 Pascal’s triangle 58
7.2 The binomial series 59
7.3 Worked problems on the binomial series 59
7.4 Further worked problems on the binomial
series 62
7.5 Practical problems involving the binomial
theorem 64
Revision Test 2 67
8 Maclaurin’s series 68
8.1 Introduction 68
8.2 Derivation of Maclaurin’s theorem 68
8.3 Conditions of Maclaurin’s series 69
8.4 Worked problems on Maclaurin’s series 69
8.5 Numerical integration using Maclaurin’s
series 73
8.6 Limiting values 74
9 Solving equations by iterative methods 77
9.1 Introduction to iterative methods 77
9.2 The bisection method 77
9.3 An algebraic method of successive
approximations 81
9.4 The Newton-Raphson method 84
10 Binary, octal and hexadecimal 87
10.1 Introduction 87
10.2 Binary numbers 87
10.3 Octal numbers 90
10.4 Hexadecimal numbers 92
Revision Test 3 96
11 Introduction to trigonometry 97
11.1 Trigonometry 97
11.2 The theorem of Pythagoras 97
11.3 Trigonometric ratios of acute angles 98
11.4 Evaluating trigonometric ratios 100
11.5 Solution of right-angled triangles 105
11.6 Angles of elevation and depression 106
11.7 Sine and cosine rules 108
11.8 Area of any triangle 108
11.9 Worked problems on the solution of
triangles and finding their areas 109
11.10 Further worked problems on solving
triangles and finding their areas 110
11.11 Practical situations involving
trigonometry 111
11.12 Further practical situations involving
trigonometry 113
12 Cartesian and polar co-ordinates 117
12.1 Introduction 117
12.2 Changing from Cartesian into polar
co-ordinates 117
12.3 Changing from polar into Cartesian
co-ordinates 119
12.4 Use of Pol/Rec functions on calculators 120
13 The circle and its properties 122
13.1 Introduction 122
13.2 Properties of circles 122
13.3 Radians and degrees 123
13.4 Arc length and area of circles and sectors 124
13.5 The equation of a circle 127
13.6 Linear and angular velocity 129
13.7 Centripetal force 130
Revision Test 4 133
14 Trigonometric waveforms 134
14.1 Graphs of trigonometric functions 134
14.2 Angles of any magnitude 135
14.3 The production of a sine and cosine wave 137
14.4 Sine and cosine curves 138
14.5 Sinusoidal form Asin(ωt ± α) 143
14.6 Harmonic synthesis with complex
waveforms 146
15 Trigonometric identities and equations 152
15.1 Trigonometric identities 152
15.2 Worked problems on trigonometric
identities 152
15.3 Trigonometric equations 154
15.4 Worked problems (i) on trigonometric
equations 154
15.5 Worked problems (ii) on trigonometric
equations 156
15.6 Worked problems (iii) on trigonometric
equations 157
15.7 Worked problems (iv) on trigonometric
equations 157
16 The relationship between trigonometric and
hyperbolic functions 159
16.1 The relationship between trigonometric
and hyperbolic functions 159
16.2 Hyperbolic identities 160
17 Compound angles 163
17.1 Compound angle formulae 163
17.2 Conversion of a sinωt +b cosωt into
R sin(ωt +α) 165
17.3 Double angles 169
17.4 Changing products of sines and cosines
into sums or differences 170
17.5 Changing sums or differences of sines and
cosines into products 171
17.6 Power waveforms in a.c. circuits 173
Revision Test 5 177
18 Functions and their curves 178
18.1 Standard curves 178
18.2 Simple transformations 181
18.3 Periodic functions 186
18.4 Continuous and discontinuous functions 186
18.5 Even and odd functions 186
18.6 Inverse functions 188
18.7 Asymptotes 190
18.8 Brief guide to curve sketching 196
18.9 Worked problems on curve sketching 197
19 Irregular areas, volumes and mean values of
waveforms 203
19.1 Areas of irregular figures 203
19.2 Volumes of irregular solids 205
19.3 The mean or average value of a waveform 206
Revision Test 6 212
20 Complex numbers 213
20.1 Cartesian complex numbers 213
20.2 The Argand diagram 214
20.3 Addition and subtraction of complex
numbers 214
20.4 Multiplication and division of complex
numbers 216
20.5 Complex equations 217
20.6 The polar form of a complex number 218
20.7 Multiplication and division in polar form 220
20.8 Applications of complex numbers 221
21 De Moivre’s theorem 225
21.1 Introduction 225
21.2 Powers of complex numbers 225
21.3 Roots of complex numbers 226
21.4 The exponential form of a complex
number 228
22 The theory of matrices and determinants 231
22.1 Matrix notation 231
22.2 Addition, subtraction and multiplication
of matrices 231
22.3 The unit matrix 235
22.4 The determinant of a 2 by 2 matrix 235
22.5 The inverse or reciprocal of a 2 by 2 matrix 236
22.6 The determinant of a 3 by 3 matrix 237
22.7 The inverse or reciprocal of a 3 by 3 matrix 239
23 The solution of simultaneous equations by
matrices and determinants 241
23.1 Solution of simultaneous equations by
matrices 241
23.2 Solution of simultaneous equations by
determinants 243
23.3 Solution of simultaneous equations using
Cramers rule 247
23.4 Solution of simultaneous equations using
the Gaussian elimination method 248
Revision Test 7 250
24 Vectors 251
24.1 Introduction 251
24.2 Scalars and vectors 251
24.3 Drawing a vector 251
24.4 Addition of vectors by drawing 252
24.5 Resolving vectors into horizontal and
vertical components 254
24.6 Addition of vectors by calculation 255
24.7 Vector subtraction 260
24.8 Relative velocity 262
24.9 i, j and k notation 263
25 Methods of adding alternating waveforms 265
25.1 Combination of two periodic functions 265
25.2 Plotting periodic functions 265
25.3 Determining resultant phasors by drawing 267
25.4 Determining resultant phasors by the sine
and cosine rules 268
25.5 Determining resultant phasors by
horizontal and vertical components 270
25.6 Determining resultant phasors by complex
numbers 272
26 Scalar and vector products 275
26.1 The unit triad 275
26.2 The scalar product of two vectors 276
26.3 Vector products 280
26.4 Vector equation of a line 283
Revision Test 8 286
27 Methods of differentiation 287
27.1 Introduction to calculus 287
27.2 The gradient of a curve 287
27.3 Differentiation from first principles 288
27.4 Differentiation of common functions 289
27.5 Differentiation of a product 292
27.6 Differentiation of a quotient 293
27.7 Function of a function 295
27.8 Successive differentiation 296
28 Some applications of differentiation 299
28.1 Rates of change 299
28.2 Velocity and acceleration 300
28.3 Turning points 303
28.4 Practical problems involving maximum
and minimum values 307
28.5 Tangents and normals 311
28.6 Small changes 312
29 Differentiation of parametric equations 315
29.1 Introduction to parametric equations 315
29.2 Some common parametric equations 315
29.3 Differentiation in parameters 315
29.4 Further worked problems on
differentiation of parametric equations 318
30 Differentiation of implicit functions 320
30.1 Implicit functions 320
30.2 Differentiating implicit functions 320
30.3 Differentiating implicit functions
containing products and quotients 321
30.4 Further implicit differentiation 322
31 Logarithmic differentiation 325
31.1 Introduction to logarithmic differentiation 325
31.2 Laws of logarithms 325
31.3 Differentiation of logarithmic functions 325
31.4 Differentiation of further logarithmic
functions 326
31.5 Differentiation of [ f (x)]
x 328
Revision Test 9 330
32 Differentiation of hyperbolic functions 331
32.1 Standard differential coefficients of
hyperbolic functions 331
32.2 Further worked problems on
differentiation of hyperbolic functions 332
33 Differentiation of inverse trigonometric and
hyperbolic functions 334
33.1 Inverse functions 334
33.2 Differentiation of inverse trigonometric
functions 334
33.3 Logarithmic forms of the inverse
hyperbolic functions 339
33.4 Differentiation of inverse hyperbolic
functions 341
34 Partial differentiation 345
34.1 Introduction to partial derivatives 345
34.2 First order partial derivatives 345
34.3 Second order partial derivatives 348
35 Total differential, rates of change and small
changes 351
35.1 Total differential 351
35.2 Rates of change 352
35.3 Small changes 354
36 Maxima, minima and saddle points for functions
of two variables 357
36.1 Functions of two independent variables 357
36.2 Maxima, minima and saddle points 358
36.3 Procedure to determine maxima, minima
and saddle points for functions of two
variables 359
36.4 Worked problems on maxima, minima
and saddle points for functions of two
variables 359
36.5 Further worked problems on maxima,
minima and saddle points for functions of
two variables 361
Revision Test 10 367
37 Standard integration 368
37.1 The process of integration 368
37.2 The general solution of integrals of the
form axn 368
37.3 Standard integrals 369
37.4 Definite integrals 372
38 Some applications of integration 375
38.1 Introduction 375
38.2 Areas under and between curves 375
38.3 Mean and r.m.s. values 377
38.4 Volumes of solids of revolution 378
38.5 Centroids 380
38.6 Theorem of Pappus 381
38.7 Second moments of area of regular
sections 383
39 Integration using algebraic substitutions 392
39.1 Introduction 392
39.2 Algebraic substitutions 392
39.3 Worked problems on integration using
algebraic substitutions 392
39.4 Further worked problems on integration
using algebraic substitutions 394
39.5 Change of limits 395
Revision Test 11 397
40 Integration using trigonometric and hyperbolic
substitutions 398
40.1 Introduction 398
40.2 Worked problems on integration of sin2 x,
cos2 x, tan2 x and cot2 x 398
40.3 Worked problems on powers of sines and
cosines 400
40.4 Worked problems on integration of
products of sines and cosines 401
40.5 Worked problems on integration using the
sin θ substitution 402
40.6 Worked problems on integration using
tan θ substitution 404
40.7 Worked problems on integration using the
sinh θ substitution 404
40.8 Worked problems on integration using the
cosh θ substitution 406
41 Integration using partial fractions 409
41.1 Introduction 409
41.2 Worked problems on integration using
partial fractions with linear factors 409
41.3 Worked problems on integration using
partial fractions with repeated linear
factors 411
41.4 Worked problems on integration using
partial fractions with quadratic factors 412
42 The t =tan
θ
2
substitution 414
42.1 Introduction 414
42.2 Worked problems on the t =tan
θ
2
substitution 415
42.3 Further worked problems on the t = tan
θ
2
substitution 416
Revision Test 12 419
43 Integration by parts 420
43.1 Introduction 420
43.2 Worked problems on integration by parts 420
43.3 Further worked problems on integration
by parts 422
44 Reduction formulae 426
44.1 Introduction 426
44.2 Using reduction formulae for integrals of
the form
xnex dx 426
44.3 Using reduction formulae for integrals of
the form
xncos x dx and
xn sin x dx 427
44.4 Using reduction formulae for integrals of
the form sinn x dx and
cosn x dx 429
44.5 Further reduction formulae 432
45 Numerical integration 435
45.1 Introduction 435
45.2 The trapezoidal rule 435
45.3 The mid-ordinate rule 437
45.4 Simpson’s rule 439
Revision Test 13 443
46 Solution of first order differential equations by
separation of variables 444
46.1 Family of curves 444
46.2 Differential equations 445
46.3 The solution of equations of the form
dy
dx = f (x) 445
46.4 The solution of equations of the form
dy
dx = f (y) 447
46.5 The solution of equations of the form
dy
dx = f (x) · f (y) 449
47 Homogeneous first order differential equations 452
47.1 Introduction 452
47.2 Procedure to solve differential equations
of the form P
dy
dx = Q 452
47.3 Worked problems on homogeneous first
order differential equations 452
47.4 Further worked problems on homogeneous
first order differential equations 454
48 Linear first order differential equations 456
48.1 Introduction 456
48.2 Procedure to solve differential equations
of the form
dy
dx + P y = Q 457
48.3 Worked problems on linear first order
differential equations 457
48.4 Further worked problems on linear first
order differential equations 458
49 Numerical methods for first order differential
equations 461
49.1 Introduction 461
49.2 Euler’s method 461
49.3 Worked problems on Euler’s method 462
49.4 An improved Euler method 466
49.5 The Runge-Kutta method 471
Revision Test 14 476
50 Second order differential equations of the form
a
d2y
dx2 +b
dy
dx +cy=0 477
50.1 Introduction 477
50.2 Procedure to solve differential equations
of the form a
d2y
dx2 +b
dy
dx +cy =0 478
50.3 Worked problems on differential equations
of the form a
d2y
dx2 +b
dy
dx +cy =0 478
50.4 Further worked problems on practical
differential equations of the form
a
d2 y
dx2 +b
dy
dx +cy =0 480
51 Second order differential equations of the form
a
d2y
dx2 +b
dy
dx +cy=f(x) 483
51.1 Complementary function and particular
integral 483
51.2 Procedure to solve differential equations
of the form a
d2y
dx2 +b
dy
dx +cy = f (x) 483
51.3 Worked problems on differential equations
of the form a
d2y
dx2 +b
dy
dx + cy = f (x)
where f (x) is a constant or polynomial 484
51.4 Worked problems on differential equations
of the form a
d2y
dx2 +b
dy
dx + cy = f (x)
where f (x) is an exponential function 486
51.5 Worked problems on differential equations
of the form a
d2y
dx2 +b
dy
dx + cy = f (x)
where f (x) is a sine or cosine function 488
51.6 Worked problems on differential equations
of the form a
d2 y
dx2 +b
dy
dx + cy = f (x)
where f (x) is a sum or a product 490
52 Power series methods of solving ordinary
differential equations 493
52.1 Introduction 493
52.2 Higher order differential coefficients as
series 493
52.3 Leibniz’s theorem 495
52.4 Power series solution by the
Leibniz–Maclaurin method 497
52.5 Power series solution by the Frobenius
method 500
52.6 Bessel’s equation and Bessel’s functions 506
52.7 Legendre’s equation and Legendre
polynomials 511
53 An introduction to partial differential equations 515
53.1 Introduction 515
53.2 Partial integration 515
53.3 Solution of partial differential equations
by direct partial integration 516
53.4 Some important engineering partial
differential equations 518
53.5 Separating the variables 518
53.6 The wave equation 519
53.7 The heat conduction equation 523
53.8 Laplace’s equation 525
Revision Test 15 528
54 Presentation of statistical data 529
54.1 Some statistical terminology 529
54.2 Presentation of ungrouped data 530
54.3 Presentation of grouped data 534
55 Measures of central tendency and dispersion 541
55.1 Measures of central tendency 541
55.2 Mean, median and mode for discrete data 541
55.3 Mean, median and mode for grouped data 542
55.4 Standard deviation 544
55.5 Quartiles, deciles and percentiles 546
56 Probability 548
56.1 Introduction to probability 548
56.2 Laws of probability 549
56.3 Worked problems on probability 549
56.4 Further worked problems on probability 551
Revision Test 16 554
57 The binomial and Poisson distributions 556
57.1 The binomial distribution 556
57.2 The Poisson distribution 559
58 The normal distribution 562
58.1 Introduction to the normal distribution 562
58.2 Testing for a normal distribution 566
59 Linear correlation 570
59.1 Introduction to linear correlation 570
59.2 The product-moment formula for
determining the linear correlation
coefficient 570
59.3 The significance of a coefficient of
correlation 571
59.4 Worked problems on linear correlation 571
60 Linear regression 575
60.1 Introduction to linear regression 575
60.2 The least-squares regression lines 575
60.3 Worked problems on linear regression 576
Revision Test 17 581
61 Introduction to Laplace transforms 582
61.1 Introduction 582
61.2 Definition of a Laplace transform 582
61.3 Linearity property of the Laplace
transform 582
61.4 Laplace transforms of elementary
functions 582
61.5 Worked problems on standard Laplace
transforms 583
62 Properties of Laplace transforms 587
62.1 The Laplace transform of eat f (t) 587
62.2 Laplace transforms of the form eat f (t) 587
62.3 The Laplace transforms of derivatives 589
62.4 The initial and final value theorems 591
63 Inverse Laplace transforms 593
63.1 Definition of the inverse Laplace transform 593
63.2 Inverse Laplace transforms of simple
functions 593
63.3 Inverse Laplace transforms using partial
fractions 596
63.4 Poles and zeros 598
64 The solution of differential equations using
Laplace transforms 600
64.1 Introduction 600
64.2 Procedure to solve differential equations
by using Laplace transforms 600
64.3 Worked problems on solving differential
equations using Laplace transforms 600
Contents xi
65 The solution of simultaneous differential
equations using Laplace transforms 605
65.1 Introduction 605
65.2 Procedure to solve simultaneous
differential equations using Laplace
transforms 605
65.3 Worked problems on solving simultaneous
differential equations by using Laplace
transforms 605
Revision Test 18 610
66 Fourier series for periodic functions of
period 2π 611
66.1 Introduction 611
66.2 Periodic functions 611
66.3 Fourier series 611
66.4 Worked problems on Fourier series of
periodic functions of period 2π 612
67 Fourier series for a non-periodic function over
range 2π 617
67.1 Expansion of non-periodic functions 617
67.2 Worked problems on Fourier series of
non-periodic functions over a range of 2π 617
68 Even and odd functions and half-range
Fourier series 623
68.1 Even and odd functions 623
68.2 Fourier cosine and Fourier sine series 623
68.3 Half-range Fourier series 626
69 Fourier series over any range 630
69.1 Expansion of a periodic function of
period L 630
69.2 Half-range Fourier series for functions
defined over range L 634
70 A numerical method of harmonic analysis 637
70.1 Introduction 637
70.2 Harmonic analysis on data given in tabular
or graphical form 637
70.3 Complex waveform considerations 641
71 The complex or exponential form of a
Fourier series 644
71.1 Introduction 644
71.2 Exponential or complex notation 644
71.3 The complex coefficients 645
71.4 Symmetry relationships 649
71.5 The frequency spectrum 652
71.6 Phasors 653
Revision Test 19 658
Essential formulae 659
Index