Mathematics Pocket Book for Engineers and Scientists

571 Pages·2019·12.388 MB·English

by  Bird, John

Why is knowledge of science
and mathematics important in
engineering?
A career in any engineering field will require both basic and advanced
mathematics and science. Without mathematics and science to determine
principles, calculate dimensions and limits, explore variations, prove con-
cepts, and so on, there would be no mobile telephones, televisions, stereo
systems, video games, microwave ovens, computers, or virtually anything
electronic. There would be no bridges, tunnels, roads, skyscrapers, automo-
biles, ships, planes, rockets or most things mechanical. There would be no
metals beyond the common ones, such as iron and copper, no plastics, no
synthetics. In fact, society would most certainly be less advanced without
the use of mathematics and science throughout the centuries and into the
future.
Electrical engineers require mathematics and science to design, develop,
test, or supervise the manufacturing and installation of electrical equip-
ment, components, or systems for commercial, industrial, military, or scien-
tific use.
Mechanical engineers require mathematics and science to perform engineer-
ing duties in planning and designing tools, engines, machines, and other
mechanically functioning equipment; they oversee installation, operation,
maintenance, and repair of such equipment as centralised heat, gas, water,
and steam systems.
Aerospace engineers require mathematics and science to perform a variety
of engineering work in designing, constructing, and testing aircraft, mis-
siles, and spacecraft; they conduct basic and applied research to evaluate
adaptability of materials and equipment to aircraft design and manufacture
and recommend improvements in testing equipment and techniques.
Nuclear engineers require mathematics and science to conduct research on
nuclear engineering problems or apply principles and theory of nuclear sci-
ence to problems concerned with release, control, and utilisation of nuclear
energy and nuclear waste disposal.
Petroleum engineers require mathematics and science to devise methods
to improve oil and gas well production and determine the need for new or
modified tool designs; they oversee drilling and offer technical advice to
achieve economical and satisfactory progress.
Industrial engineers require mathematics and science to design, develop,
test, and evaluate integrated systems for managing industrial production
processes, including human work factors, quality control, inventory control,
logistics and material flow, cost analysis, and production coordination.
Environmental engineers require mathematics and science to design, plan,
or perform engineering duties in the prevention, control, and remediation
of environmental health hazards, using various engineering disciplines;
their work may include waste treatment, site remediation, or pollution con-
trol technology.
Civil engineers require mathematics and science in all levels in civil engi-
neering – structural engineering, hydraulics and geotechnical engineering
are all fields that employ mathematical tools such as differential equations,
tensor analysis, field theory, numerical methods and operations research.
Architects require knowledge of algebra, geometry, trigonometry and calcu-
lus. They use mathematics for several reasons, leaving aside the necessary
use of mathematics in the engineering of buildings. Architects use geometry
because it defines the spatial form of a building, and they use mathemat-
ics to design forms that are considered beautiful or harmonious. The front
cover of this text shows a modern London architecture and the financial
district, all of which at some stage required in its design a knowledge of
mathematics.
Knowledge of mathematics and science is clearly needed by each of the dis-
ciplines listed above.
It is intended that this text – Mathematics Pocket Book for Engineers and
Scientists – will provide a step by step, helpful reference, to essential mathe-
matics topics needed by engineers and scientists.
Mathematics Pocket Book for
Engineers and Scientists
Fifth Edition
John Bird
Fifth edition published 2020
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
and by Routledge
52 Vanderbilt Avenue, New York, NY 10017
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2020 John Bird
The right of John Bird to be identified as author of this work has been asserted by him in
accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form
or by any electronic, mechanical, or other means, now known or hereafter invented, i ncluding
photocopying and recording, or in any information storage or retrieval system, without
permission in writing from the publishers.
Trademark notice: Product or corporate names may be trademarks or registered trademarks,
and are used only for identification and explanation without intent to infringe.
First edition published as Newnes Mathematics for Engineers Pocket Book by Newnes 1983
Fourth edition published as Engineering Mathematics Pocket Book by Routledge 2008
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalog record has been requested for this book
ISBN: 978-0-367-26653-0 (hbk)
ISBN: 978-0-367-26652-3 (pbk)
ISBN: 978-0-429-29440-2 (ebk)
Typeset in Frutiger 45 Light by
Servis FIlmsetting Ltd, Stockport, Cheshire.
Mathematics Pocket Book for
Engineers and Scientists
John Bird is the former Head of Applied Electronics in the Faculty of Technology at
Highbury College, Portsmouth, UK. More recently, he has combined freelance lec-
turing at the University of Portsmouth, with examiner responsibilities for Advanced
Mathematics with City and Guilds and examining for International Baccalaureate.
He has over 45 years’ experience of successfully teaching, lecturing, instructing,
training, educating and planning trainee engineers study programmes. He is the
author of 140 textbooks on engineering, science and mathematical subjects, with
worldwide sales of over one million copies. He is a chartered engineer, a chartered
mathematician, a chartered scientist and a Fellow of three professional institu-
tions. He is currently lecturing at the Defence College of Marine Engineering in the
Defence College of Technical Training at H.M.S. Sultan, Gosport, Hampshire, UK,
one of the largest technical training establishments in Europe.

Contents
Preface xiii
Section 1 Engineering conversions, constants and symbols 1
Chapter 1 General conversions and the Greek alphabet 2
Chapter 2 Basic SI units, derived units and common prefixes 3
Chapter 3 Some physical and mathematical constants 5
Chapter 4 Recommended mathematical symbols 7
Chapter 5 Symbols for physical quantities 9
Section 2 Some algebra topics 17
Chapter 6 Introduction to algebra 18
Chapter 7 Polynomial division 22
Chapter 8 The factor theorem 23
Chapter 9 The remainder theorem 25
Chapter 10 Continued fractions 25
Chapter 11 Solving simple equations 27
Chapter 12 Transposing formulae 32
Chapter 13 Solving simultaneous equations 36
Chapter 14 Solving quadratic equations by factorising 42
Chapter 15 Solving quadratic equations by completing the square 45
Chapter 16 Solving quadratic equations by formula 47
Chapter 17 Logarithms 49
Chapter 18 Exponential functions 53
Chapter 19 Napierian logarithms 55
Chapter 20 Hyperbolic functions 58
Chapter 21 Partial fractions 63
Section 3 Some number topics 67
Chapter 22 Simple number sequences 68
Chapter 23 Arithmetic progressions 70
Chapter 24 Geometric progressions 71
Chapter 25 Inequalities 73
Chapter 26 The binomial series 80
Chapter 27 Maclaurin’s theorem 84
Chapter 28 Limiting values – L’Hopital’s rule 87
Chapter 29 S olving equations by iterative methods (1) – the
bisection method 88
Chapter 30 S olving equations by iterative methods (2) – an
algebraic method of successive approximations 90
Chapter 31 S olving equations by iterative methods (3) – the
Newton-Raphson method 92
Chapter 32 Computer numbering systems 93
viii Contents
Section 4 Areas and volumes 101
Chapter 33 Area of plane figures 103
Chapter 34 Circles 106
Chapter 35 Volumes and surface areas of regular solids 111
Chapter 36 Volumes and surface areas of frusta of pyramids and cones 116
Chapter 37 The frustum and zone of a sphere 119
Chapter 38 Areas and volumes of irregular figures and solids 122
Chapter 39 The mean or average value of a waveform 127
Section 5 Geometry and trigonometry 131
Chapter 40 Types and properties of angles 133
Chapter 41 Properties of triangles 133
Chapter 42 The theorem of Pythagoras 135
Chapter 43 Trigonometric ratios of acute angles 136
Chapter 44 Evaluating trigonometric ratios 137
Chapter 45 Fractional and surd forms of trigonometric ratios 138
Chapter 46 Solution of right-angled triangles 139
Chapter 47 Cartesian and polar co-ordinates 142
Chapter 48 Sine and cosine rules and areas of any triangle 144
Chapter 49 Graphs of trigonometric functions 149
Chapter 50 Angles of any magnitude 149
Chapter 51 Sine and cosine waveforms 152
Chapter 52 Trigonometric identities and equations 157
Chapter 53 T he relationship between trigonometric and hyperbolic
functions 161
Chapter 54 Compound angles 163
Section 6 Graphs 170
Chapter 55 The straight-line graph 172
Chapter 56 Determination of law 174
Chapter 57 Graphs with logarithmic scales 179
Chapter 58 Graphical solution of simultaneous equations 183
Chapter 59 Quadratic graphs 184
Chapter 60 Graphical solution of cubic equations 189
Chapter 61 Polar curves 190
Chapter 62 The ellipse and hyperbola 196
Chapter 63 Graphical functions 197
Section 7 Complex numbers 204
Chapter 64 General complex number formulae 205
Chapter 65 Cartesian form of a complex number 205
Chapter 66 Polar form of a complex number 208
Chapter 67 Applications of complex numbers 210
Chapter 68 De Moivre’s theorem 212
Chapter 69 Exponential form of a complex number 214
Section 8 Vectors 216
Chapter 70 Scalars and vectors 217
Chapter 71 Vector addition 217
Chapter 72 Resolution of vectors 219
Chapter 73 Vector subtraction 220
Chapter 74 Relative velocity 222
Chapter 75 i, j, k notation 224
Contents ix
Chapter 76 Combination of two periodic functions 225
Chapter 77 The scalar product of two vectors 227
Chapter 78 Vector products 230
Section 9 Matrices and determinants 233
Chapter 79 Addition, subtraction and multiplication of matrices 234
Chapter 80 The determinant and inverse of a 2 by 2 matrix 235
Chapter 81 The determinant of a 3 by 3 matrix 236
Chapter 82 The inverse of a 3 by 3 matrix 238
Chapter 83 Solution of simultaneous equations by matrices 239
Chapter 84 Solution of simultaneous equations by determinants 242
Chapter 85 S olution of simultaneous equations using
Cramer’s rule 245
Chapter 86 S olution of simultaneous equations using Gaussian
elimination 247
Chapter 87 Eigenvalues and eigenvectors 248
Section 10 Boolean algebra and logic circuits 253
Chapter 88 Boolean algebra and switching circuits 254
Chapter 89 Simplifying Boolean expressions 257
Chapter 90 Laws and rules of Boolean algebra 258
Chapter 91 De Morgan’s laws 259
Chapter 92 Karnaugh maps 261
Chapter 93 Logic circuits and gates 266
Chapter 94 Universal logic gates 269
Section 11 Differential calculus and its applications 273
Chapter 95 Common standard derivatives 275
Chapter 96 Products and quotients 276
Chapter 97 Function of a function 277
Chapter 98 Successive differentiation 278
Chapter 99 Differentiation of hyperbolic functions 279
Chapter 100 Rates of change using differentiation 280
Chapter 101 Velocity and acceleration 281
Chapter 102 Turning points 282
Chapter 103 Tangents and normals 285
Chapter 104 Small changes using differentiation 287
Chapter 105 Parametric equations 288
Chapter 106 Differentiating implicit functions 291
Chapter 107 Differentiation of logarithmic functions 293
Chapter 108 Differentiation of inverse trigonometric functions 295
Chapter 109 Differentiation of inverse hyperbolic functions 298
Chapter 110 Partial differentiation 302
Chapter 111 Total differential 305
Chapter 112 Rates of change using partial differentiation 306
Chapter 113 Small changes using partial differentiation 307
Chapter 114 Maxima, minima and saddle points of functions of
two variables 308
Section 12 Integral calculus and its applications 315
Chapter 115 Standard integrals 317
Chapter 116 Non-standard integrals 320
Chapter 117 Integration using algebraic substitutions 321