Higher Engineering Mathematics, Fifth Edition

Cover......Page 1
Higher Engineering Mathematics......Page 4
Copyright page - ISBN: 9780750681520......Page 5
Contents......Page 6
Preface......Page 16
Syllabus guidance......Page 18
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 Simple inequalities......Page 31
2.3 Inequalities involving a modulus......Page 32
2.4 Inequalities involving quotients......Page 33
2.5 Inequalities involving square functions......Page 34
2.6 Quadratic inequalities......Page 35
3.2 Worked problems on partial fractions with linear factors......Page 37
3.3 Worked problems on partial fractions with repeated linear factors......Page 40
3.4 Worked problems on partial fractions with quadratic factors......Page 41
4.2 Laws of logarithms......Page 43
4.3 Indicial equations......Page 45
4.4 Graphs of logarithmic functions......Page 46
4.5 The exponential function......Page 47
4.6 The power series for e^x......Page 48
4.7 Graphs of exponential functions......Page 50
4.8 Napierian logarithms......Page 52
4.9 Laws of growth and decay......Page 54
4.10 Reduction of exponential laws to linear form......Page 57
5.1 Introduction to hyperbolic functions......Page 60
5.2 Graphs of hyperbolic functions......Page 62
5.3 Hyperbolic identities......Page 63
5.4 Solving equations involving hyperbolic functions......Page 66
5.5 Series expansions for cosh x and sinh x......Page 67
Assignment 1......Page 69
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 80
7.5 Practical problems involving the binomial theorem......Page 83
8.3 Conditions of Maclaurin's series......Page 86
8.4 Worked problems on Maclaurin's series......Page 87
8.5 Numerical integration using Maclaurin's series......Page 90
8.6 Limiting values......Page 91
Assignment 2......Page 94
9.2 The bisection method......Page 95
9.3 An algebraic method of successive approximations......Page 99
9.4 The Newton-Raphson method......Page 102
10.2 Conversion of binary to denary......Page 105
10.3 Conversion of denary to binary......Page 106
10.4 Conversion of denary to binary via octal......Page 107
10.5 Hexadecimal numbers......Page 109
11.1 Boolean algebra and switching circuits......Page 113
11.3 Laws and rules of Boolean algebra......Page 118
11.4 De Morgan's laws......Page 120
11.5 Karnaugh maps......Page 121
11.6 Logic circuits......Page 125
11.7 Universal logic gates......Page 129
Assignment 3......Page 133
12.2 The theorem of Pythagoras......Page 134
12.3 Trigonometric ratios of acute angles......Page 135
12.4 Solution of right-angled triangles......Page 137
12.5 Angles of elevation and depression......Page 138
12.6 Evaluating trigonometric ratios......Page 140
12.7 Sine and cosine rules......Page 143
12.9 Worked problems on the solution of triangles and finding their areas......Page 144
12.10 Further worked problems on solving triangles and finding their areas......Page 145
12.11 Practical situations involving trigonometry......Page 147
12.12 Further practical situations involving trigonometry......Page 149
13.2 Changing from Cartesian into polar co-ordinates......Page 152
13.3 Changing from polar into Cartesian co-ordinates......Page 154
13.4 Use of R -> P and P -> R functions on calculators......Page 155
14.2 Properties of circles......Page 156
14.3 Arc length and area of a sector......Page 157
14.4 Worked problems on arc length and sector of a circle......Page 158
14.5 The equation of a circle......Page 159
14.6 Linear and angular velocity......Page 161
14.7 Centripetal force......Page 163
Assignment 4......Page 165
15.2 Angles of any magnitude......Page 167
15.3 The production of a sine and cosine wave......Page 170
15.4 Sine and cosine curves......Page 171
15.5 Sinusoidal form A sin (ωt ± α)......Page 176
15.6 Harmonic synthesis with complex waveforms......Page 179
16.2 Worked problems on trigonometric identities......Page 185
16.3 Trigonometric equations......Page 186
16.4 Worked problems (i) on trigonometric equations......Page 187
16.5 Worked problems (ii) on trigonometric equations......Page 188
16.6 Worked problems (iii) on trigonometric equations......Page 189
16.7 Worked problems (iv) on trigonometric equations......Page 190
17.1 The relationship between trigonometric and hyperbolic functions......Page 192
17.2 Hyperbolic identities......Page 193
18.1 Compound angle formulae......Page 195
18.2 Conversion of sin ωt + b cos ωt into R sin(ωt + α)......Page 197
18.3 Double angles......Page 201
18.4 Changing products of sines and cosines into sums or differences......Page 202
18.5 Changing sums or differences of sines and cosines into products......Page 203
18.6 Power waveforms in a.c. circuits......Page 204
Assignment 5......Page 208
19.1 Standard curves......Page 210
19.2 Simple transformations......Page 213
19.5 Even and odd functions......Page 218
19.6 Inverse functions......Page 220
19.7 Asymptotes......Page 222
19.8 Brief guide to curve sketching......Page 228
19.9 Worked problems on curve sketching......Page 229
20.1 Areas of irregular figures......Page 235
20.2 Volumes of irregular solids......Page 237
20.3 The mean or average value of a waveform......Page 238
21.2 Vector addition......Page 244
21.3 Resolution of vectors......Page 246
21.4 Vector subtraction......Page 248
21.5 Relative velocity......Page 250
21.6 Combination of two periodic functions......Page 251
22.1 The unit triad......Page 256
22.2 The scalar product of two vectors......Page 257
22.3 Vector products......Page 260
22.4 Vector equation of a line......Page 264
Assignment 6......Page 266
23.1 Cartesian complex numbers......Page 268
23.3 Addition and subtraction of complex numbers......Page 269
23.4 Multiplication and division of complex numbers......Page 270
23.5 Complex equations......Page 272
23.6 The polar form of a complex number......Page 273
23.7 Multiplication and division in polar form......Page 275
23.8 Applications of complex numbers......Page 276
24.2 Powers of complex numbers......Page 280
24.3 Roots of complex numbers......Page 281
24.4 The exponential form of a complex number......Page 283
25.2 Addition, subtraction and multiplication of matrices......Page 286
25.4 The determinant of a 2 by 2 matrix......Page 290
25.5 The inverse or reciprocal of a 2 by 2 matrix......Page 291
25.6 The determinant of a 3 by 3 matrix......Page 292
25.7 The inverse or reciprocal of a 3 by 3 matrix......Page 293
26.1 Solution of simultaneous equations by matrices......Page 296
26.2 Solution of simultaneous equations by determinants......Page 298
26.3 Solution of simultaneous equations using Cramers rule......Page 302
26.4 Solution of simultaneous equations using the Gaussian elimination method......Page 303
Assignment 7......Page 305
27.1 The gradient of a curve......Page 306
27.3 Differentiation of common functions......Page 307
27.4 Differentiation of a product......Page 311
27.5 Differentiation of a quotient......Page 312
27.6 Function of a function......Page 314
27.7 Successive differentiation......Page 315
28.1 Rates of change......Page 317
28.2 Velocity and acceleration......Page 318
28.3 Turning points......Page 321
28.4 Practical problems involving maximum and minimum values......Page 325
28.5 Tangents and normals......Page 329
28.6 Small changes......Page 330
29.3 Differentiation in parameters......Page 333
29.4 Further worked problems on differentiation of parametric equations......Page 335
30.2 Differentiating implicit functions......Page 338
30.3 Differentiating implicit functions containing products and quotients......Page 339
30.4 Further implicit differentiation......Page 340
31.3 Differentiation of logarithmic functions......Page 343
31.4 Differentiation of [ƒ(x)]^x......Page 346
Assignment 8......Page 348
32.1 Standard differential coefficients of hyperbolic functions......Page 349
32.2 Further worked problems on differentiation of hyperbolic functions......Page 350
33.2 Differentiation of inverse trigonometric functions......Page 351
33.3 Logarithmic forms of the inverse hyperbolic functions......Page 356
33.4 Differentiation of inverse hyperbolic functions......Page 357
34.2 First order partial derivatives......Page 362
34.3 Second order partial derivatives......Page 365
35.1 Total differential......Page 368
35.2 Rates of change......Page 369
35.3 Small changes......Page 371
36.2 Maxima, minima and saddle points......Page 374
36.3 Procedure to determine maxima, minima and saddle points for functions of two variables......Page 375
36.4 Worked problems on maxima, minima and saddle points for functions of two variables......Page 376
36.5 Further worked problems on maxima, minima and saddle points for functions of two variables......Page 378
Assignment 9......Page 384
37.3 Standard integrals......Page 386
37.4 Definite integrals......Page 390
38.2 Areas under and between curves......Page 393
38.3 Mean and r.m.s. values......Page 395
38.4 Volumes of solids of revolution......Page 396
38.5 Centroids......Page 397
38.6 Theorem of Pappus......Page 399
38.7 Second moments of area of regular sections......Page 401
39.3 Worked problems on integration using algebraic substitutions......Page 410
39.5 Change of limits......Page 412
Assignment 10......Page 415
40.2 Worked problems on integration of (sin x)^2, (cos x)^2, (tan x)^2, and (cot x)^2......Page 416
40.3 Worked problems on powers of sines and cosines......Page 418
40.4 Worked problems on integration of products of sines and cosines......Page 419
40.5 Worked problems on integration using the sin θ substitution......Page 420
40.7 Worked problems on integration using the sinh θ substitution......Page 422
40.8 Worked problems on integration using the cosh θ substitution......Page 424
41.2 Worked problems on integration using partial fractions with linear factors......Page 427
41.3 Worked problems on integration using partial fractions with repeated linear factors......Page 428
41.4 Worked problems on integration using partial fractions with quadratic factors......Page 429
42.2 Worked problems on the t = tan(θ/2) substitution......Page 432
42.3 Further worked problems on the t = tan(θ/2) substitution......Page 434
Assignment 11......Page 436
43.2 Worked problems on integration by parts......Page 437
43.3 Further worked problems on integration by parts......Page 439
44.2 Using reduction formulae for integrals of the form \int x^n e^x dx......Page 443
44.3 Using reduction formulae for integrals of the form \int x^n cos x dx and \int x^n sin x dx......Page 444
44.4 Using reduction formulae for integrals of the form \int (sin x)^n dx and \int (cos x)^n dx......Page 446
44.5 Further reduction formulae......Page 449
45.2 The trapezoidal rule......Page 452
45.3 The mid-ordinate rule......Page 454
45.4 Simpson's rule......Page 456
Assignment 12......Page 460
46.1 Family of curves......Page 462
46.3 The solution of equations of the form dy/dx = ƒ(x)......Page 463
46.4 The solution of equations of the form dy/dx = ƒ(x)......Page 465
46.5 The solution of equations of the form dy/dx = ƒ(x)·ƒ(y)......Page 467
47.3 Worked problems on homogeneous first order differential equations......Page 470
47.4 Further worked problems on homogeneous first order differential equations......Page 471
48.2 Procedure to solve differential equations of the form dy/dx + Py = Q......Page 474
48.3 Worked problems on linear first order differential equations......Page 475
48.4 Further worked problems on linear first order differential equations......Page 476
49.2 Euler's method......Page 479
49.3 Worked problems on Euler's method......Page 480
49.4 An improved Euler method......Page 484
49.5 The Runge-Kutta method......Page 488
Assignment 13......Page 493
50.2 Procedure to solve differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0......Page 494
50.3 Worked problems on differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0......Page 495
50.4 Further worked problems on practical differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = 0......Page 497
51.2 Procedure to solve differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = f(x)......Page 500
51.3 Worked problems on differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = f(x)......Page 501
51.4 Worked problems on differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = f(x)......Page 503
51.5 Worked problems on differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = f(x)......Page 505
51.6 Worked problems on differential equations of the form α\frac{d^2 y}{dx^2} + b\frac{dy}{dx} + cy = f(x)......Page 507
52.2 Higher order differential coefficients as series......Page 510
52.3 Leibniz's theorem......Page 512
52.4 Power series solution by the Leibniz–Maclaurin method......Page 514
52.5 Power series solution by the Frobenius method......Page 517
52.6 Bessel's equation and Bessel's functions......Page 523
52.7 Legendre's equation and Legendre polynomials......Page 528
53.2 Partial integration......Page 531
53.3 Solution of partial differential equations by direct partial integration......Page 532
53.5 Separating the variables......Page 534
53.6 The wave equation......Page 535
53.7 The heat conduction equation......Page 539
53.8 Laplace's equation......Page 541
Assignment 14......Page 544
54.1 Some statistical terminology......Page 546
54.2 Presentation of ungrouped data......Page 547
54.3 Presentation of grouped data......Page 551
55.2 Mean, median and mode for discrete data......Page 557
55.3 Mean, median and mode for grouped data......Page 558
55.4 Standard deviation......Page 560
55.5 Quartiles, deciles and percentiles......Page 562
56.2 Laws of probability......Page 564
56.3 Worked problems on probability......Page 565
56.4 Further worked problems on probability......Page 567
Assignment 15......Page 570
57.1 The binomial distribution......Page 572
57.2 The Poisson distribution......Page 575
58.1 Introduction to the normal distribution......Page 578
58.2 Testing for a normal distribution......Page 582
59.2 The product-moment formula for determining the linear correlation coefficient......Page 586
59.4 Worked problems on linear correlation......Page 587
60.2 The least-squares regression lines......Page 590
60.3 Worked problems on linear regression......Page 591
Assignment 16......Page 595
61.3 The sampling distribution of the means......Page 596
61.4 The estimation of population parameters based on a large sample size......Page 600
61.5 Estimating the mean of a population based on a small sample size......Page 605
62.2 Type I and Type II errors......Page 609
62.3 Significance tests for population means......Page 616
62.4 Comparing two sample means......Page 621
63.1 Chi-square values......Page 626
63.2 Fitting data to theoretical distributions......Page 627
63.3 Introduction to distribution-free tests......Page 632
63.4 The sign test......Page 633
63.5 Wilcoxon signed-rank test......Page 635
63.6 The Mann-Whitney test......Page 639
Assignment 17......Page 644
64.4 Laplace transforms of elementary functions......Page 646
64.5 Worked problems on standard Laplace transforms......Page 648
65.2 Laplace transforms of the form e^{αt}ƒ(t)......Page 651
65.3 The Laplace transforms of derivatives......Page 653
65.4 The initial and final value theorems......Page 655
66.2 Inverse Laplace transforms of simple functions......Page 657
66.3 Inverse Laplace transforms using partial fractions......Page 659
66.4 Poles and zeros......Page 661
67.3 Worked problems on solving differential equations using Laplace transforms......Page 664
68.3 Worked problems on solving simultaneous differential equations by using Laplace transforms......Page 669
Assignment 18......Page 674
69.3 Fourier series......Page 676
69.4 Worked problems on Fourier series of periodic functions of period 2π......Page 677
70.2 Worked problems on Fourier series of non-periodic functions over a range of 2π......Page 682
71.2 Fourier cosine and Fourier sine series......Page 688
71.3 Half-range Fourier series......Page 691
72.1 Expansion of a periodic function of period L......Page 695
72.2 Half-range Fourier series for functions defined over range L......Page 699
73.2 Harmonic analysis on data given in tabular or graphical form......Page 702
73.3 Complex waveform considerations......Page 705
74.2 Exponential or complex notation......Page 709
74.3 Complex coefficients......Page 710
74.4 Symmetry relationships......Page 714
74.5 The frequency spectrum......Page 717
74.6 Phasors......Page 718
Assignment 19......Page 723
Essential formulae......Page 724
C......Page 740
E......Page 741
I......Page 742
P......Page 743
S......Page 744
Z......Page 745