Table Of Content

Engineering Mathematics Pocket Book Fourth Edition John Bird This page intentionally left blank Engineering Mathematics Pocket Book Fourth edition John Bird BSc(Hons), CEng, CSci, CMath, FIMA, FIET, MIEE, FIIE, FCollT AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Newnes is an imprint of Elsevier Newnes is an imprint of Elsevier Linace House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First published as the Newnes Mathematics for Engineers Pocket Book 1983 Reprinted 1988, 1990 (twice), 1991, 1992, 1993 Second edition 1997 Third edition as the Newnes Engineering Mathematics Pocket Book 2001 Fourth edition as the Engineering Mathematics Pocket Book 2008 Copyright © 2008 John Bird, Published by Elsevier Ltd. All rights reserved The right of John Bird to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permission may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone ((cid:2)44) (0) 1865 843830; fax ((cid:2)44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier website at http:elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN: 978-0-7506-8153-7 For information on all Newnes publications visit our web site at http://books.elsevier.com Typeset by Charon Tec Ltd., A Macmillan Company. (www.macmillansolutions.com) Printed and bound in United Kingdom 08 09 10 11 12 10 9 8 7 6 5 4 3 2 1 Contents Preface xi 1 Engineering Conversions, Constants and Symbols 1 1.1 General conversions 1 1.2 Greek alphabet 2 1.3 Basic SI units, derived units and common prefixes 3 1.4 Some physical and mathematical constants 5 1.5 Recommended mathematical symbols 7 1.6 Symbols for physical quantities 10 2 Some Algebra Topics 20 2.1 Polynomial division 20 2.2 The factor theorem 21 2.3 The remainder theorem 23 2.4 Continued fractions 24 2.5 Solution of quadratic equations by formula 25 2.6 Logarithms 28 2.7 Exponential functions 31 2.8 Napierian logarithms 32 2.9 Hyperbolic functions 36 2.10 Partial fractions 41 3 Some Number Topics 46 3.1 Arithmetic progressions 46 3.2 Geometric progressions 47 3.3 The binomial series 49 3.4 Maclaurin’s theorem 54 3.5 Limiting values 57 3.6 Solving equations by iterative methods 58 3.7 Computer numbering systems 65 4 Areas and Volumes 73 4.1 Area of plane figures 73 4.2 Circles 77 4.3 Volumes and surface areas of regular solids 82 4.4 Volumes and surface areas of frusta of pyramids and cones 88 vi Contents 4.5 The frustum and zone of a sphere 92 4.6 Areas and volumes of irregular figures and solids 95 4.7 The mean or average value of a waveform 101 5 Geometry and Trigonometry 105 5.1 Types and properties of angles 105 5.2 Properties of triangles 106 5.3 Introduction to trigonometry 108 5.4 Trigonometric ratios of acute angles 109 5.5 Evaluating trigonometric ratios 110 5.6 Fractional and surd forms of trigonometric ratios 112 5.7 Solution of right-angled triangles 113 5.8 Cartesian and polar co-ordinates 116 5.9 Sine and cosine rules and areas of any triangle 119 5.10 Graphs of trigonometric functions 124 5.11 Angles of any magnitude 125 5.12 Sine and cosine waveforms 127 5.13 Trigonometric identities and equations 134 5.14 The relationship between trigonometric and hyperbolic functions 139 5.15 Compound angles 141 6 Graphs 149 6.1 The straight line graph 149 6.2 Determination of law 152 6.3 Logarithmic scales 158 6.4 Graphical solution of simultaneous equations 163 6.5 Quadratic graphs 164 6.6 Graphical solution of cubic equations 170 6.7 Polar curves 171 6.8 The ellipse and hyperbola 178 6.9 Graphical functions 180 7 Vectors 188 7.1 Scalars and vectors 188 7.2 Vector addition 189 7.3 Resolution of vectors 191 7.4 Vector subtraction 192 7.5 Relative velocity 195 7.6 Combination of two periodic functions 197 7.7 The scalar product of two vectors 200 7.8 Vector products 203 8 Complex Numbers 206 8.1 General formulae 206 8.2 Cartesian form 206 Contents vii 8.3 Polar form 209 8.4 Applications of complex numbers 211 8.5 De Moivre’s theorem 213 8.6 Exponential form 215 9 Matrices and Determinants 217 9.1 Addition, subtraction and multiplication of matrices 217 9.2 The determinant and inverse of a 2 by 2 matrix 218 9.3 The determinant of a 3 by 3 matrix 220 9.4 The inverse of a 3 by 3 matrix 221 9.5 Solution of simultaneous equations by matrices 223 9.6 Solution of simultaneous equations by determinants 226 9.7 Solution of simultaneous equations using Cramer’s rule 230 9.8 Solution of simultaneous equations using Gaussian elimination 232 10 Boolean Algebra and Logic Circuits 234 10.1 Boolean algebra and switching circuits 234 10.2 Simplifying Boolean expressions 238 10.3 Laws and rules of Boolean algebra 239 10.4 De Morgan’s laws 241 10.5 Karnaugh maps 242 10.6 Logic circuits and gates 248 10.7 Universal logic gates 253 11 Differential Calculus and its Applications 258 11.1 Common standard derivatives 258 11.2 Products and quotients 259 11.3 Function of a function 261 11.4 Successive differentiation 262 11.5 Differentiation of hyperbolic functions 263 11.6 Rates of change using differentiation 264 11.7 Velocity and acceleration 265 11.8 Turning points 267 11.9 Tangents and normals 270 11.10 Small changes using differentiation 272 11.11 Parametric equations 273 11.12 Differentiating implicit functions 276 11.13 Differentiation of logarithmic functions 279 11.14 Differentiation of inverse trigonometric functions 281 11.15 Differentiation of inverse hyperbolic functions 284 11.16 Partial differentiation 289 11.17 Total differential 292 11.18 Rates of change using partial differentiation 293 11.19 Small changes using partial differentiation 294 11.20 Maxima, minima and saddle points of functions of two variables 295 viii Contents 12 Integral Calculus and its Applications 303 12.1 Standard integrals 303 12.2 Non-standard integrals 307 12.3 Integration using algebraic substitutions 307 12.4 Integration using trigonometric and hyperbolic substitutions 310 12.5 Integration using partial fractions 317 θ 12.6 The t(cid:3)tan substitution 319 2 12.7 Integration by parts 323 12.8 Reduction formulae 326 12.9 Numerical integration 331 12.10 Area under and between curves 336 12.11 Mean or average values 343 12.12 Root mean square values 345 12.13 Volumes of solids of revolution 347 12.14 Centroids 350 12.15 Theorem of Pappus 354 12.16 Second moments of area 359 13 Differential Equations 366 dy 13.1 The solution of equations of the form (cid:3)f(x) 366 dx dy 13.2 The solution of equations of the form (cid:3)f(y) 367 dx dy 13.3 The solution of equations of the form (cid:3)f(x).f(y) 368 dx 13.4 Homogeneous first order differential equations 371 13.5 Linear first order differential equations 373 13.6 Second order differential equations of the form d2y dy a (cid:2)b (cid:2)cy(cid:3)0 375 dx2 dx 13.7 Second order differential equations of the form d2y dy a (cid:2)b (cid:2)cy(cid:3)f(x) 379 dx2 dx 13.8 Numerical methods for first order differential equations 385 13.9 Power series methods of solving ordinary differential equations 394 13.10 Solution of partial differential equations 405 14 Statistics and Probability 416 14.1 Presentation of ungrouped data 416 14.2 Presentation of grouped data 420 Contents ix 14.3 Measures of central tendency 424 14.4 Quartiles, deciles and percentiles 429 14.5 Probability 431 14.6 The binomial distribution 434 14.7 The Poisson distribution 435 14.8 The normal distribution 437 14.9 Linear correlation 443 14.10 Linear regression 445 14.11 Sampling and estimation theories 447 14.12 Chi-square values 454 14.13 The sign test 457 14.14 Wilcoxon signed-rank test 460 14.15 The Mann-Whitney test 464 15 Laplace Transforms 472 15.1 Standard Laplace transforms 472 15.2 Initial and final value theorems 477 15.3 Inverse Laplace transforms 480 15.4 Solving differential equations using Laplace transforms 483 15.5 Solving simultaneous differential equations using Laplace transforms 487 16 Fourier Series 492 16.1 Fourier series for periodic functions of period 2π 492 16.2 Fourier series for a non-periodic function over range 2π 496 16.3 Even and odd functions 498 16.4 Half range Fourier series 501 16.5 Expansion of a periodic function of period L 504 16.6 Half-range Fourier series for functions defined over range L 508 16.7 The complex or exponential form of a Fourier series 511 16.8 A numerical method of harmonic analysis 518 16.9 Complex waveform considerations 522 Index 525