Higher engineering mathematics

Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 14
Syllabus Guidance......Page 16
1.2 Revision of basic laws......Page 20
1.3 Revision of equations......Page 22
1.4 Polynomial division......Page 25
1.5 The factor theorem......Page 27
1.6 The remainder theorem......Page 29
2.2 Worked problems on partial
fractions with linear factors......Page 32
2.3 Worked problems on partial fractions with repeated linear
factors......Page 35
2.4 Worked problems on partial
fractions with quadratic factors......Page 36
3.1 Introduction to logarithms......Page 39
3.2 Laws of logarithms......Page 41
3.3 Indicial equations......Page 43
3.4 Graphs of logarithmic functions......Page 44
4.1 Introduction to exponential
functions......Page 46
4.2 The power series for ex......Page 47
4.3 Graphs of exponential functions......Page 48
4.4 Napierian logarithms......Page 50
4.5 Laws of growth and decay......Page 53
4.6 Reduction of exponential laws to
linear form......Page 56
Revision Test 1......Page 59
5.1 Introduction to hyperbolic
functions......Page 60
5.2 Graphs of hyperbolic functions......Page 62
5.3 Hyperbolic identities......Page 64
5.4 Solving equations involving
hyperbolic functions......Page 66
5.5 Series expansions for cosh x and sinh x......Page 68
6.2 Worked problems on arithmetic
progressions......Page 70
6.3 Further worked problems on
arithmetic progressions......Page 71
6.4 Geometric progressions......Page 73
6.5 Worked problems on geometric
progressions......Page 74
6.6 Further worked problems on
geometric progressions......Page 75
7.1 Pascal’s triangle......Page 77
7.3 Worked problems on the
binomial series......Page 78
7.4 Further worked problems on the binomial series......Page 81
7.5 Practical problems involving the binomial theorem......Page 83
Revision Test 2......Page 86
8.2 Derivationof Maclaurin’s theorem......Page 87
8.4 Worked problems on
Maclaurin’s series......Page 88
8.5 Numerical integration using
Maclaurin’s series......Page 92
8.6 Limiting values......Page 93
9.2 The bisection
method......Page 96
9.3 An algebraic method of successive
approximations......Page 100
9.4 The Newton-Raphson
method......Page 103
10.2 Binary numbers......Page 106
10.3 Octal numbers......Page 109
10.4 Hexadecimal numbers......Page 111
Revision Test 3......Page 115
11.2 The theorem
of Pythagoras......Page 116
11.3 Trigonometric ratios of acute
angles......Page 117
11.4 Evaluating trigonometric ratios......Page 119
11.5 Solution of right-angled
triangles......Page 124
11.6 Angles of elevation and
depression......Page 125
11.8 Area of any triangle......Page 127
11.9 Worked problems on the solution of triangles and
finding their areas......Page 128
11.10 Further worked problems on solving triangles and finding
their areas......Page 129
11.11 Practical situations involving
trigonometry......Page 130
11.12 Further practical situation
sinvolving trigonometry......Page 132
12.2 Changing from
Cartesian into polar co-ordinates......Page 136
12.3 Changing from
polar into Cartesian co-ordinates......Page 138
12.4 Use of Pol/Rec functions on
calculators......Page 139
13.2 Properties of circles......Page 141
13.3 Radians and degrees......Page 142
13.4 Arc length and area of circles
and sectors......Page 143
13.5 The equation of a circle......Page 146
13.6 Linear and angular velocity......Page 148
13.7 Centripetal force......Page 149
Revision Test 4......Page 152
14.1 Graphs of trigonometric
functions......Page 153
14.2 Angles of any magnitude......Page 154
14.3 The production of a sine and
cosine wave......Page 156
14.4 Sine and cosine curves......Page 157
14.5 Sinusoidal form A sin(ωt ± α)......Page 162
14.6 Harmonic synthesis with
complex waveforms......Page 165
15.2 Worked problems on
trigonometric identities......Page 171
15.4 Worked problems (i) on trigonometric equations......Page 173
15.5 Worked problems (ii) on trigonometric equations......Page 175
15.7 Worked problems (iv) on trigonometric equations......Page 176
16.1 The relationship between trigonometric and hyperbolic
functions......Page 178
16.2 Hyperbolic identities......Page 179
17.1 Compound angle formulae......Page 182
17.2 Conversion of a sin ωt + b cos ωt into R sin(ωt + α)......Page 184
17.3 Double angles......Page 188
17.4 Changing products of sines and cosines into sums or differences......Page 189
17.5 Changing sums or differences of
sines and cosines into products......Page 190
17.6 Power waveforms in a.c. circuits......Page 192
Revision Test 5......Page 196
18.1 Standard curves......Page 197
18.2 Simple transformations......Page 200
18.5 Even and odd functions......Page 205
18.6 Inverse functions......Page 207
18.7 Asymptotes......Page 209
18.8 Brief guide to curve sketching......Page 215
18.9 Worked problems on curve
sketching......Page 216
19.1 Areas of irregular figures......Page 222
19.2 Volumes of irregular solids......Page 224
19.3 The mean or average value of a waveform......Page 225
Revision Test 6......Page 231
20.1 Cartesian complex numbers......Page 232
20.3 Addition and subtraction of
complex numbers......Page 233
20.4 Multiplication and division of
complex numbers......Page 235
20.5 Complex equations......Page 236
20.6 The polar form of a complex
number......Page 237
20.7 Multiplication and division in
polar form......Page 239
20.8 Applications of complex
numbers......Page 240
21.2 Powers of complex numbers......Page 244
21.3 Roots of complex numbers......Page 245
21.4 The exponential formof a
complex number......Page 247
22.2 Addition, subtraction and
multiplication of matrices......Page 250
22.4 The determinant of a 2 by 2
matrix......Page 254
22.5 The inverse or reciprocal of a 2
by 2 matrix......Page 255
22.6 The determinant of a 3 by 3
matrix......Page 256
22.7 The inverse or reciprocal of a 3 by 3
matrix......Page 258
23.1 Solution of simultaneous equations by
matrices......Page 260
23.2 Solution of simultaneous
equations by determinants......Page 262
23.3 Solution of simultaneous
equations using Cramers rule......Page 266
23.4 Solution of simultaneous equations using the Gaussian
elimination method......Page 267
Revision Test 7......Page 269
24.3 Drawing a vector......Page 270
24.4 Addition of vectors by drawing......Page 271
24.5 Resolving vectors into horizontal
and vertical components......Page 273
24.6 Addition of vectors by
calculation......Page 274
24.7 Vector subtraction......Page 279
24.8 Relative velocity......Page 281
24.9 i, j and k notation......Page 282
25.2 Plotting periodic functions......Page 284
25.3 Determining resultant phasors
by drawing......Page 286
25.4 Determining resultant phasors
by the sine and cosine rules......Page 287
25.5 Determining resultant phasors by horizontal and vertical
components......Page 289
25.6 Determining resultant phasors
by complex numbers......Page 291
26.1 The unit triad......Page 294
26.2 The scalar product of two
vectors......Page 295
26.3 Vector products......Page 299
26.4 Vector equation of a line......Page 302
Revision Test 8......Page 305
27.2 The gradient of a curve......Page 306
27.3 Differentiation from first
principles......Page 307
27.4 Differentiation of common
functions......Page 308
27.5 Differentiation of a product......Page 311
27.6 Differentiation of a quotient......Page 312
27.7 Function of a function......Page 314
27.8 Successive differentiation......Page 315
28.1 Rates of change......Page 318
28.2 Velocity and acceleration......Page 319
28.3 Turning points......Page 322
28.4 Practical problems involving
maximum and minimum values......Page 326
28.5 Tangents and normals......Page 330
28.6 Small changes......Page 331
29.3 Differentiation in parameters......Page 334
29.4 Further worked problems on
differentiation of parametric equations......Page 337
30.2 Differentiating implicit
functions......Page 339
30.3 Differentiating implicit functions containing products
and quotients......Page 340
30.4 Further implicit differentiation......Page 341
31.3 Differentiation of logarithmic
functions......Page 344
31.4 Differentiation of further
logarithmic functions......Page 345
31.5 Differentiation of [f(x)]x
......Page 347
Revision Test 9......Page 349
32.1 Standard differential coefficients of hyperbolic functions......Page 350
32.2 Further worked problems on differentiation of hyperbolic functions......Page 351
33.2 Differentiation of inverse
trigonometric functions......Page 353
33.3 Logarithmic forms of inverse
hyperbolic functions......Page 358
33.4 Differentiation of inverse
hyperbolic functions......Page 360
34.2 First order partial derivatives......Page 364
34.3 Second order partial derivatives......Page 367
35.1 Total differential......Page 370
35.2 Rates of change......Page 371
35.3 Small changes......Page 373
36.1 Functions of two independent
variables......Page 376
36.2 Maxima,minima and saddle
points......Page 377
36.4 Worked problems on maxima, minima and saddle points for functions of two variables......Page 378
36.5 Further worked problems on maxima, minima and saddle points for functions of two variables......Page 380
Revision Test 10......Page 386
37.2 The general solution of integrals 
of the form axn......Page 387
37.3 Standard integrals......Page 388
37.4 Definite integrals......Page 391
38.2 Areas under and between curves......Page 394
38.3 Mean and r.m.s. values......Page 396
38.4 Volumes of solids of revolution......Page 397
38.5 Centroids......Page 399
38.6 Theorem of Pappus......Page 400
38.7 Second moments of area of regular sections......Page 402
39.3 Worked problems on integration using algebraic substitutions......Page 411
39.4 Further worked problems on integration using algebraic substitutions......Page 413
39.5 Change of limits......Page 414
Revision Test 11......Page 416
40.2 Worked problems on integration of sin2 x, cos2 x, tan2 x and cot2 x
......Page 417
40.3 Worked problems on powers of sines and cosines......Page 419
40.4 Worked problems on integration of products of sines and cosines......Page 420
40.5 Worked problems on integration using the sin θ substitution......Page 421
40.7 Worked problems on integration using the sinh θ substitution......Page 423
40.8 Worked problems on integration using the cosh θ substitution......Page 425
41.2 Worked problems on integration using partial fractions with linear factors......Page 428
41.3 Worked problems on integration using partial fractions with repeated linear factors......Page 430
41.4 Worked problems on integration using partial fractions with quadratic factors......Page 431
42.1 Introduction......Page 433
42.2 Worked problems on the t = tan θ/2 substitution......Page 434
42.3 Further worked problems on the t = tan θ/2 substitution......Page 435
Revision Test 12......Page 438
43.2 Worked problems on integration
by parts......Page 439
43.3 Further worked problems on integration by parts......Page 441
44.2 Using reduction formulae for integrals of the form ?xn ex dx......Page 445
44.3 Using reduction formulae for integrals of the form ?xn cosx dx and ? xn sinx  dx......Page 446
44.4 Using reduction formulae for integrals of the form ?sinn x dx and ?cosn x dx......Page 448
44.5 Further reduction formulae......Page 451
45.2 The trapezoidal rule......Page 454
45.3 The mid-ordinate rule......Page 456
45.4 Simpson’s rule
......Page 458
Revision Test 13......Page 462
46.1 Family of curves......Page 463
46.3 The solution of equations of the form dy/dx = f(x)......Page 464
46.4 The solution of equations of the form dy/dx = f(y)......Page 466
46.5 The solution of equations of the form dy/dx = f(x).f(y)......Page 468
47.3 Worked problems on homogeneous first order differential equations......Page 471
47.4 Further worked problems on homogeneous first order differential equations......Page 473
48.1 Introduction......Page 475
48.3 Worked problems on linear first order differential equations......Page 476
48.4 Further worked problems on linear first order differential equations......Page 477
49.2 Euler’s method......Page 480
49.3 Worked problems on Euler’s method......Page 481
49.4 An improved Euler method......Page 485
49.5 The Runge-Kutta method......Page 490
Revision Test 14......Page 495
50.1 Introduction......Page 496
50.3 Worked problems on differential equations of the form a d2y/dx2 + b dy/dx + cy = 0......Page 497
50.4 Further worked problems on practical differential equations of the form a d2y/dx2 + b dy/dx + cy = 0......Page 499
51.2 Procedure to solve differentialequations of the form a d2y/dx2 +b dy/dx +cy = f(x)
......Page 502
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
......Page 503
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
......Page 505
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
......Page 507
51.6 Worked problems on differential equations of the form a d2y/dx2 +b dy/dx +cy = f(x) where f (x) is a sum or a product
......Page 509
52.2 Higher order differential coefficients as series......Page 512
52.3 Leibniz’s theorem......Page 514
52.4 Power series solution by the Leibniz–Maclaurin method......Page 516
52.5 Power series solution by the 
Frobenius method......Page 519
52.6 Bessel’s equation and Bessel’s functions......Page 525
52.7 Legendre’s equation and Legendre polynomials......Page 530
53.2 Partial integration......Page 534
53.3 Solution of partial differential equations by direct partial integration......Page 535
53.5 Separating the variables......Page 537
53.6 The wave equation......Page 538
53.7 The heat conduction equation......Page 542
53.8 Laplace’s equation......Page 544
Revision Test 15......Page 547
54.1 Some statistical terminology......Page 548
54.2 Presentation of ungrouped data......Page 549
54.3 Presentation of grouped data......Page 553
55.2 Mean, median andmode for discrete data......Page 560
55.3 Mean, median andmode for grouped data......Page 561
55.4 Standard deviation......Page 563
55.5 Quartiles, deciles and percentiles......Page 565
56.1 Introduction to probability......Page 567
56.3 Worked problemson probability......Page 568
56.4 Further worked problems on probability......Page 570
Revision Test 16......Page 573
57.1 The binomial distribution......Page 575
57.2 The Poisson distribution......Page 578
58.1 Introduction to the normal distribution......Page 581
58.2 Testing for a normal distribution......Page 585
59.2 The product-moment formula for determining the linear correlation coefficient......Page 589
59.4 Worked problems on linear correlation......Page 590
60.2 The least-squares regression lines......Page 594
60.3 Worked problems on linear regression......Page 595
Revision Test 17......Page 600
61.4 Laplace transforms of elementary functions......Page 601
61.5 Worked problems on standard Laplace transforms......Page 602
62.2 Laplace transforms of the form eat f(t)
......Page 606
62.3 The Laplace transforms of derivatives......Page 608
62.4 The initial and final value theorems......Page 610
63.2 Inverse Laplace transforms of 
simple functions......Page 612
63.3 Inverse Laplace transforms using 
partial fractions......Page 615
63.4 Poles and zeros......Page 617
64.3 Worked problems on solving differential equations using Laplace transforms......Page 619
65.3 Worked problems on solving simultaneous differential equations by using Laplace transforms......Page 624
Revision Test 18......Page 629
66.3 Fourier series......Page 630
66.4 Worked problems on Fourier series of periodic functions of period 2π......Page 631
67.2 Worked problems on Fourier series of non-periodic functions over a range of 2π......Page 636
68.2 Fourier cosine and Fourier sine series......Page 642
68.3 Half-range Fourier series......Page 645
69.1 Expansion of a periodic function of period L......Page 649
69.2 Half-range Fourier series for functions defined over range L......Page 653
70.2 Harmonic analysis on data given in tabular or graphical form......Page 656
70.3 Complex waveform considerations......Page 660
71.2 Exponential or complexnotation......Page 663
71.3 The complex coefficients......Page 664
71.4 Symmetry relationships......Page 668
71.5 The frequency spectrum......Page 671
71.6 Phasors......Page 672
Revision Test 19......Page 677
Essential formulae......Page 678
Index......Page 694