Anthony Croft is Professor of Mathematics Education at Loughborough University. Robert Davison was formerly Head of Quality at the Faculty of Technology; De Montfort University. Martin Hargreaves is a Chartered Physicist. James Flint is Reader in Wireless Systems Engineering at Loughborough University.

## Table of contents

Table of contents :

Cover……Page 1

Title Page……Page 4

Copyright Page……Page 5

Dedication……Page 6

Contents……Page 8

Preface……Page 18

Acknowledgements……Page 20

1.1 Introduction……Page 22

1.2 Laws of indices……Page 23

1.3 Number bases……Page 32

1.4 Polynomial equations……Page 41

1.5 Algebraic fractions……Page 47

1.6 Solution of inequalities……Page 54

1.7 Partial fractions……Page 60

1.8 Summation notation……Page 67

Review exercises 1……Page 71

2.1 Introduction……Page 75

2.2 Numbers and intervals……Page 76

2.3 Basic concepts of functions……Page 77

2.4 Review of some common engineering functions and techniques……Page 91

Review exercises 2……Page 134

3.1 Introduction……Page 136

3.3 The trigonometric ratios……Page 137

3.4 The sine, cosine and tangent functions……Page 141

3.5 The sinc x function……Page 144

3.6 Trigonometric identities……Page 146

3.7 Modelling waves using sin t and cos t……Page 152

3.8 Trigonometric equations……Page 165

Review exercises 3……Page 171

4.2 Cartesian coordinate system – two dimensions……Page 175

4.3 Cartesian coordinate system – three dimensions……Page 178

4.4 Polar coordinates……Page 180

4.5 Some simple polar curves……Page 184

4.6 Cylindrical polar coordinates……Page 187

4.7 Spherical polar coordinates……Page 191

Review exercises 4……Page 194

5.2 Set theory……Page 196

5.3 Logic……Page 204

5.4 Boolean algebra……Page 206

Review exercises 5……Page 218

6.1 Introduction……Page 221

6.2 Sequences……Page 222

6.3 Series……Page 230

6.4 The binomial theorem……Page 235

6.5 Power series……Page 239

6.6 Sequences arising from the iterative solution of non-linear equations……Page 240

Review exercises 6……Page 243

7.2 Vectors and scalars: basic concepts……Page 245

7.3 Cartesian components……Page 253

7.4 Scalar fields and vector fields……Page 261

7.5 The scalar product……Page 262

7.6 The vector product……Page 267

7.7 Vectors of n dimensions……Page 274

Review exercises 7……Page 276

8.1 Introduction……Page 278

8.2 Basic definitions……Page 279

8.3 Addition, subtraction and multiplication……Page 280

8.4 Using matrices in the translation and rotation of vectors……Page 288

8.5 Some special matrices……Page 292

8.6 The inverse of a 2 x 2 matrix……Page 295

8.7 Determinants……Page 299

8.8 The inverse of a 3 x 3 matrix……Page 302

8.9 Application to the solution of simultaneous equations……Page 304

8.10 Gaussian elimination……Page 307

8.11 Eigenvalues and eigenvectors……Page 315

8.12 Analysis of electrical networks……Page 328

8.13 Iterative techniques for the solution of simultaneous equations……Page 333

8.14 Computer solutions of matrix problems……Page 340

Review exercises 8……Page 342

9.1 Introduction……Page 345

9.2 Complex numbers……Page 346

9.3 Operations with complex numbers……Page 349

9.4 Graphical representation of complex numbers……Page 353

9.5 Polar form of a complex number……Page 354

9.6 Vectors and complex numbers……Page 357

9.7 The exponential form of a complex number……Page 358

9.8 Phasors……Page 361

9.9 De Moivre’s theorem……Page 365

9.10 Loci and regions of the complex plane……Page 372

Review exercises 9……Page 375

10.1 Introduction……Page 377

10.2 Graphical approach to differentiation……Page 378

10.3 Limits and continuity……Page 379

10.4 Rate of change at a specific point……Page 383

10.5 Rate of change at a general point……Page 385

10.6 Existence of derivatives……Page 391

10.7 Common derivatives……Page 393

10.8 Differentiation as a linear operator……Page 396

Review exercises 10……Page 406

11.2 Rules of differentiation……Page 407

11.3 Parametric, implicit and logarithmic differentiation……Page 414

11.4 Higher derivatives……Page 421

Review exercises 11……Page 425

12.2 Maximum points and minimum points……Page 427

12.3 Points of inflexion……Page 436

12.4 The Newton–Raphson method for solving equations……Page 439

12.5 Differentiation of vectors……Page 444

Review exercises 12……Page 448

13.1 Introduction……Page 449

13.2 Elementary integration……Page 450

13.3 Definite and indefinite integrals……Page 463

Review exercises 13……Page 474

14.2 Integration by parts……Page 478

14.3 Integration by substitution……Page 484

14.4 Integration using partial fractions……Page 487

Review exercises 14……Page 489

15.2 Average value of a function……Page 492

15.3 Root mean square value of a function……Page 496

Review exercises 15……Page 500

16.2 Orthogonal functions……Page 501

16.3 Improper integrals……Page 504

16.4 Integral properties of the delta function……Page 510

16.5 Integration of piecewise continuous functions……Page 512

16.6 Integration of vectors……Page 514

Review exercises 16……Page 515

17.2 Trapezium rule……Page 517

17.3 Simpson’s rule……Page 521

Review exercises 17……Page 526

18.1 Introduction……Page 528

18.2 Linearization using first-order Taylor polynomials……Page 529

18.3 Second-order Taylor polynomials……Page 534

18.4 Taylor polynomials of the nth order……Page 538

18.5 Taylor’s formula and the remainder term……Page 542

18.6 Taylor and Maclaurin series……Page 545

Review exercises 18……Page 553

19.1 Introduction……Page 555

19.2 Basic definitions……Page 556

19.3 First-order equations: simple equations and separation of variables……Page 561

19.4 First-order linear equations: use of an integrating factor……Page 568

19.5 Second-order linear equations……Page 579

19.6 Constant coefficient equations……Page 581

19.7 Series solution of differential equations……Page 605

19.8 Bessel’s equation and Bessel functions……Page 608

Review exercises 19……Page 622

20.2 Analogue simulation……Page 624

20.3 Higher order equations……Page 627

20.4 State-space models……Page 630

20.5 Numerical methods……Page 636

20.6 Euler’s method……Page 637

20.7 Improved Euler method……Page 641

20.8 Runge–Kutta method of order 4……Page 644

Review exercises 20……Page 647

21.1 Introduction……Page 648

21.2 Definition of the Laplace transform……Page 649

21.3 Laplace transforms of some common functions……Page 650

21.4 Properties of the Laplace transform……Page 652

21.5 Laplace transform of derivatives and integrals……Page 656

21.6 Inverse Laplace transforms……Page 659

21.7 Using partial fractions to find the inverse Laplace transform……Page 662

21.8 Finding the inverse Laplace transform using complex numbers……Page 664

21.9 The convolution theorem……Page 668

21.10 Solving linear constant coefficient differential equations using the Laplace transform……Page 670

21.11 Transfer functions……Page 680

21.12 Poles, zeros and the s plane……Page 689

21.13 Laplace transforms of some special functions……Page 696

Review exercises 21……Page 699

22.1 Introduction……Page 702

22.2 Basic definitions……Page 703

22.3 Rewriting difference equations……Page 707

22.4 Block diagram representation of difference equations……Page 709

22.5 Design of a discrete-time controller……Page 714

22.6 Numerical solution of difference equations……Page 716

22.7 Definition of the z transform……Page 719

22.8 Sampling a continuous signal……Page 723

22.9 The relationship between the z transform and the Laplace transform……Page 725

22.10 Properties of the z transform……Page 730

22.11 Inversion of z transforms……Page 736

22.12 The z transform and difference equations……Page 739

Review exercises 22……Page 741

23.1 Introduction……Page 743

23.2 Periodic waveforms……Page 744

23.3 Odd and even functions……Page 747

23.4 Orthogonality relations and other useful identities……Page 753

23.5 Fourier series……Page 754

23.6 Half-range series……Page 766

23.7 Parseval’s theorem……Page 769

23.8 Complex notation……Page 770

23.9 Frequency response of a linear system……Page 772

Review exercises 23……Page 776

24.1 Introduction……Page 778

24.2 The Fourier transform – definitions……Page 779

24.3 Some properties of the Fourier transform……Page 782

24.4 Spectra……Page 787

24.5 The t-ω duality principle……Page 789

24.6 Fourier transforms of some special functions……Page 791

24.7 The relationship between the Fourier transform and the Laplace transform……Page 793

24.8 Convolution and correlation……Page 795

24.9 The discrete Fourier transform……Page 804

24.10 Derivation of the d.f.t…….Page 808

24.11 Using the d.f.t. to estimate a Fourier transform……Page 811

24.12 Matrix representation of the d.f.t…….Page 813

24.13 Some properties of the d.f.t…….Page 814

24.14 The discrete cosine transform……Page 816

24.15 Discrete convolution and correlation……Page 822

Review exercises 24……Page 842

25.2 Functions of more than one variable……Page 844

25.3 Partial derivatives……Page 846

25.4 Higher order derivatives……Page 850

25.5 Partial differential equations……Page 853

25.6 Taylor polynomials and Taylor series in two variables……Page 856

25.7 Maximum and minimum points of a function of two variables……Page 862

Review exercises 25……Page 867

26.2 Partial differentiation of vectors……Page 870

26.3 The gradient of a scalar field……Page 872

26.4 The divergence of a vector field……Page 877

26.5 The curl of a vector field……Page 880

26.6 Combining the operators grad, div and curl……Page 882

26.7 Vector calculus and electromagnetism……Page 885

Review exercises 26……Page 886

27.2 Line integrals……Page 888

27.3 Evaluation of line integrals in two dimensions……Page 892

27.4 Evaluation of line integrals in three dimensions……Page 894

27.5 Conservative fields and potential functions……Page 896

27.6 Double and triple integrals……Page 901

27.7 Some simple volume and surface integrals……Page 910

27.8 The divergence theorem and Stokes’ theorem……Page 916

27.9 Maxwell’s equations in integral form……Page 920

Review exercises 27……Page 922

28.1 Introduction……Page 924

28.2 Introducing probability……Page 925

28.3 Mutually exclusive events: the addition law of probability……Page 930

28.4 Complementary events……Page 934

28.5 Concepts from communication theory……Page 936

28.6 Conditional probability: the multiplication law……Page 940

28.7 Independent events……Page 946

Review exercises 28……Page 951

29.1 Introduction……Page 954

29.2 Random variables……Page 955

29.3 Probability distributions – discrete variable……Page 956

29.4 Probability density functions – continuous variable……Page 957

29.5 Mean value……Page 959

29.6 Standard deviation……Page 962

29.7 Expected value of a random variable……Page 964

29.8 Standard deviation of a random variable……Page 967

29.9 Permutations and combinations……Page 969

29.10 The binomial distribution……Page 974

29.11 The Poisson distribution……Page 978

29.12 The uniform distribution……Page 982

29.13 The exponential distribution……Page 983

29.14 The normal distribution……Page 984

29.15 Reliability engineering……Page 991

Review exercises 29……Page 998

Appendix I Representing a continuous function and a sequence as a sum of weighted impulses……Page 1000

Appendix II The Greek alphabet……Page 1002

Appendix IV The binomial expansion of (n-N/n)n……Page 1003

A……Page 1004

C……Page 1005

D……Page 1008

E……Page 1009

F……Page 1011

H……Page 1012

I……Page 1013

L……Page 1014

M……Page 1016

P……Page 1017

R……Page 1019

S……Page 1020

T……Page 1022

V……Page 1023

X……Page 1024

Z……Page 1025