engineering mathematics pocket book fourth edition pdf

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 (ϩ44) (0) 1865 843830; fax (ϩ44) (0) 1865 853333; email: permissions@elsevier.com 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 Contents Preface xi 1.1 1.2 1.3 1.4 1.5 1.6 Engineering Conversions, Constants and Symbols General conversions Greek alphabet Basic SI units, derived units and common prefixes Some physical and mathematical constants Recommended mathematical symbols Symbols for physical quantities 1 10 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Some Algebra Topics Polynomial division The factor theorem The remainder theorem Continued fractions Solution of quadratic equations by formula Logarithms Exponential functions Napierian logarithms Hyperbolic functions Partial fractions 20 20 21 23 24 25 28 31 32 36 41 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Some Number Topics Arithmetic progressions Geometric progressions The binomial series Maclaurin’s theorem Limiting values Solving equations by iterative methods Computer numbering systems 46 46 47 49 54 57 58 65 4.1 4.2 4.3 4.4 Areas and Volumes Area of plane figures Circles Volumes and surface areas of regular solids Volumes and surface areas of frusta of pyramids and cones 73 73 77 82 88 vi Contents 4.5 4.6 4.7 The frustum and zone of a sphere Areas and volumes of irregular figures and solids The mean or average value of a waveform 92 95 101 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 105 105 106 108 109 110 112 113 116 119 124 125 127 134 5.15 Geometry and Trigonometry Types and properties of angles Properties of triangles Introduction to trigonometry Trigonometric ratios of acute angles Evaluating trigonometric ratios Fractional and surd forms of trigonometric ratios Solution of right-angled triangles Cartesian and polar co-ordinates Sine and cosine rules and areas of any triangle Graphs of trigonometric functions Angles of any magnitude Sine and cosine waveforms Trigonometric identities and equations The relationship between trigonometric and hyperbolic functions Compound angles 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Graphs The straight line graph Determination of law Logarithmic scales Graphical solution of simultaneous equations Quadratic graphs Graphical solution of cubic equations Polar curves The ellipse and hyperbola Graphical functions 149 149 152 158 163 164 170 171 178 180 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Vectors Scalars and vectors Vector addition Resolution of vectors Vector subtraction Relative velocity Combination of two periodic functions The scalar product of two vectors Vector products 188 188 189 191 192 195 197 200 203 8.1 8.2 Complex Numbers General formulae Cartesian form 206 206 206 139 141 Contents vii 8.3 8.4 8.5 8.6 Polar form Applications of complex numbers De Moivre’s theorem Exponential form 209 211 213 215 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Matrices and Determinants Addition, subtraction and multiplication of matrices The determinant and inverse of a by matrix The determinant of a by matrix The inverse of a by matrix Solution of simultaneous equations by matrices Solution of simultaneous equations by determinants Solution of simultaneous equations using Cramer’s rule Solution of simultaneous equations using Gaussian elimination 217 217 218 220 221 223 226 230 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 Boolean Algebra and Logic Circuits Boolean algebra and switching circuits Simplifying Boolean expressions Laws and rules of Boolean algebra De Morgan’s laws Karnaugh maps Logic circuits and gates Universal logic gates 234 234 238 239 241 242 248 253 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 Differential Calculus and its Applications Common standard derivatives Products and quotients Function of a function Successive differentiation Differentiation of hyperbolic functions Rates of change using differentiation Velocity and acceleration Turning points Tangents and normals Small changes using differentiation Parametric equations Differentiating implicit functions Differentiation of logarithmic functions Differentiation of inverse trigonometric functions Differentiation of inverse hyperbolic functions Partial differentiation Total differential Rates of change using partial differentiation Small changes using partial differentiation Maxima, minima and saddle points of functions of two variables 258 258 259 261 262 263 264 265 267 270 272 273 276 279 281 284 289 292 293 294 232 295 viii Contents 12 12.1 12.2 12.3 12.4 Integral Calculus and its Applications Standard integrals Non-standard integrals Integration using algebraic substitutions Integration using trigonometric and hyperbolic substitutions Integration using partial fractions θ The t ϭ tan substitution Integration by parts Reduction formulae Numerical integration Area under and between curves Mean or average values Root mean square values Volumes of solids of revolution Centroids Theorem of Pappus Second moments of area 303 303 307 307 Differential Equations dy The solution of equations of the form ϭ f(x) dx dy 13.2 The solution of equations of the form ϭ f(y) dx dy ϭ f(x).f(y) 13.3 The solution of equations of the form dx 13.4 Homogeneous first order differential equations 13.5 Linear first order differential equations 13.6 Second order differential equations of the form d2y dy a ϩb ϩ cy ϭ dx dx 13.7 Second order differential equations of the form d2y dy a ϩb ϩ cy ϭ f(x) dx dx 13.8 Numerical methods for first order differential equations 13.9 Power series methods of solving ordinary differential equations 13.10 Solution of partial differential equations 366 366 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15 12.16 13 13.1 14 14.1 14.2 Statistics and Probability Presentation of ungrouped data Presentation of grouped data 310 317 319 323 326 331 336 343 345 347 350 354 359 367 368 371 373 375 379 385 394 405 416 416 420 Contents ix 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 Measures of central tendency Quartiles, deciles and percentiles Probability The binomial distribution The Poisson distribution The normal distribution Linear correlation Linear regression Sampling and estimation theories Chi-square values The sign test Wilcoxon signed-rank test The Mann-Whitney test 424 429 431 434 435 437 443 445 447 454 457 460 464 15 15.1 15.2 15.3 15.4 15.5 Laplace Transforms Standard Laplace transforms Initial and final value theorems Inverse Laplace transforms Solving differential equations using Laplace transforms Solving simultaneous differential equations using Laplace transforms 472 472 477 480 483 Fourier Series Fourier series for periodic functions of period 2π Fourier series for a non-periodic function over range 2π Even and odd functions Half range Fourier series Expansion of a periodic function of period L Half-range Fourier series for functions defined over range L The complex or exponential form of a Fourier series A numerical method of harmonic analysis Complex waveform considerations 492 492 496 498 501 504 508 511 518 522 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 Index 487 525 Fourier Series 517 which is demonstrated below (1 Ϫ cos nπ) From equation (17), cn ϭ Ϫj nπ j4 2 When n ϭ 1, c1 ϭ Ϫj (1 Ϫ cos π) ϭ Ϫj (1 Ϫ Ϫ1) ϭ Ϫ (1)π π π When n ϭ 2, c2 ϭ Ϫj (1 Ϫ cos 2π) ϭ ; in fact, all even values of 2π cn will be zero j4 2 When n ϭ 3, c3 ϭ Ϫj (1 Ϫ cos 3π) ϭ Ϫj (1 Ϫ Ϫ1) ϭ Ϫ 3π 3π 3π j4 j4 , c7 ϭ Ϫ By similar reasoning, c5 ϭ Ϫ , and so on 5π 7π When n ϭ Ϫ1, cϪ1 ϭ Ϫj When n ϭ Ϫ3, cϪ3 ϭϪ j j4 2 (1Ϫ cos (Ϫπ)) ϭ ϩj π (1ϪϪ1) ϭ ϩ π (Ϫ1)π j4 2 (1Ϫ cos(Ϫ3π) ) ϭ ϩj (1ϪϪ1) ϭϩ (Ϫ3)π 3π 3π By similar reasoning, cϪ5 ϭ ϩ j4 j4 , cϪ7 ϭ ϩ , and so on 5π 7π Since the waveform is odd, c0 ϭ a0 ϭ From equation (18), f( x ) ϭ ϱ ∑ Ϫj nπ (1Ϫ cos nπ) e jnx nϭϪϱ Hence, f( x ) ϭ Ϫ j4 jx j4 j3x j4 j5x j4 j7x e Ϫ e Ϫ e Ϫ e Ϫ… π 3π 5π 7π j4 Ϫjx j4 Ϫj3x j4 Ϫj5x j4 Ϫj7x ϩ ϩ ϩ ϩ… e ϩ e e e π 3π 5π 7π ⎞ ⎛ j4 3x ⎛ j4 j4 j4 Ϫ3x ⎞⎟ ϭ ⎜⎜Ϫ e jx ϩ e Ϫjx ⎟⎟⎟ ϩ ⎜⎜Ϫ e ϩ e ⎟⎟ ⎜⎝ π ⎜ ⎠ ⎝ 3π ⎠ π 3π ⎛ j4 5x j4 Ϫ5x ⎞⎟ ⎜ ϩ ⎜Ϫ e ϩ e ⎟⎟ ϩ ⎜⎝ 5π ⎠ 5π j4 j4 jx j4 (e Ϫ eϪjx ) Ϫ 3π (e3x Ϫ eϪ3x ) Ϫ 5π (e5x Ϫ eϪ5x ) ϩ π jx 3x 5x (e Ϫ eϪjx ) ϩ (e Ϫ eϪ3x ) ϩ (e Ϫ eϪ5x ) ϩ ϭ j5π jπ j3π by multiplying top and bottom by j ϭϪ 518 Engineering Mathematics Pocket Book ⎛⎜ e jx Ϫ eϪjx ⎞⎟ ⎛⎜ e j3x Ϫ eϪj3 ⎞⎟ ⎛⎜ e j5x Ϫ eϪj5x ⎞⎟ ⎟⎟ ϩ ⎟⎟ ϩ ⎟⎟ ⎜⎜ ⎜⎜ ⎜ ⎟ ⎟ ⎟⎠ π⎝ 2j 2j 2j ⎠ 3π ⎝ ⎠ 5π ⎜⎝ ϩ by rearranging 8 ϭ sin x ϩ sin 3x ϩ sin 5x ϩ π 3π 3x from equation 10, page 511 ϭ i.e f(x) ‫؍‬ Hence, f(x) ‫؍‬ ⎞ ⎛⎜ 1 ⎜ sin x ؉ sin 3x ؉ sin 5x ؉ sin 7x ؉ ⎟⎟⎟ ⎠ π ⎜⎝ ؕ ∑ n‫؍‬؊ؕ ≡ ؊j (1 ؊ cos nπ) e jnx nπ ⎞ ⎛⎜ 1 ⎜ sin x ؉ sin 3x ؉ sin 5x ؉ sin 7x ؉ ⎟⎟⎟ ⎠ π ⎜⎝ 16.8 A numerical method of harmonic analysis Many practical waveforms can be represented by simple mathematical expressions, and, by using Fourier series, the magnitude of their harmonic components determined, as above For waveforms not in this category, analysis may be achieved by numerical methods Harmonic analysis is the process of resolving a periodic, nonsinusoidal quantity into a series of sinusoidal components of ascending order of frequency The trapezoidal rule can be used to evaluate the Fourier coefficients, which are given by: p an ≈ bn ≈ P P p ∑ yk (19) ∑ yk cos nxk (20) a0 ≈ k‫؍‬1 p k‫؍‬1 p ∑ yk k‫؍‬1 sin nxk (21) Fourier Series 519 Application: A graph of voltage V against angle θ is shown in Figure 16.13 Determine a Fourier series to represent the graph Voltage (volts) y10 80 60 40 y y2 20 Ϫ20 Ϫ40 Ϫ60 Ϫ80 y9 90 y11 y12 y8 180 270 y7 y3 y4 y5 y6 360 ␪ degrees Figure 16.13 The values of the ordinates y1, y2, y3, … are 62, 35, Ϫ38, Ϫ64, Ϫ63, Ϫ52, Ϫ28, 24, 80, 96, 90 and 70, the 12 equal intervals each being of width 30° (If a larger number of intervals are used, results having a greater accuracy are achieved) The voltage may be analysed into its first three constituent components as follows: The data is tabulated in the proforma shown in Table 16.1 p From equation (19), a0 Ϸ 1 ∑ yk ϭ 12 (212) ϭ 17.67 (since p ϭ 12) p kϭ1 From equation (20), an Ϸ ∑ yk cos nxk p kϭ1 p ( 417.94 ) ϭ 69.66, 12 a2 Ϸ (Ϫ39) ϭ Ϫ6.50 12 hence a1 Ϸ and a3 Ϸ (Ϫ49) ϭ Ϫ8.17 12 b3 Ϸ (55) ϭ 9.17 12 p From equation (21), bn Ϸ hence ∑ yk sin nxk p kϭ1 (Ϫ278.53) ϭ Ϫ46.42 , 12 b2 Ϸ (29.43) ϭ 4.91 and 12 b1 Ϸ Table 16.1 Ordinates θ V cos θ V cos θ sin θ V sin θ Y1 30 62 0.866 53.69 0.5 31 Y2 60 35 0.5 17.5 0.866 30.31 cos 2θ V cos 2θ 0.5 Ϫ0.5 sin 2θ V sin 2θ 31 0.866 53.69 0 62 Ϫ17.5 0.866 30.31 Ϫ1 Ϫ35 0 0 Ϫ1 38 Ϫ64 0 0 Ϫ63 Ϫ1 52 0 28 Y3 90 Ϫ38 0 Ϫ38 Ϫ1 38 Y4 120 Ϫ64 Ϫ0.5 32 0.866 Ϫ55.42 Ϫ0.5 32 Ϫ0.866 55.42 Ϫ31.5 0.5 Ϫ31.5 Ϫ0.866 54.56 Ϫ52 Y5 150 Ϫ63 Ϫ0.866 54.56 0.5 Y6 180 Ϫ52 Ϫ1 52 Y7 210 Ϫ28 Ϫ0.866 Y8 240 24 Ϫ0.5 Y9 270 80 24.25 Ϫ12 0 Ϫ0.5 0.5 Ϫ14 0.866 Ϫ24.25 0 Ϫ1 Ϫ12 0.866 20.78 24 0 Ϫ1 Ϫ80 Ϫ1 Ϫ80 0 80 Ϫ0.5 Ϫ48 Ϫ0.866 Ϫ83.14 Ϫ1 Ϫ96 0 0.5 45 Ϫ0.866 Ϫ77.94 0 Ϫ1 Ϫ90 70 70 0 300 96 0.5 48 Ϫ0.866 Ϫ83.14 90 0.866 77.94 Ϫ0.5 Ϫ45 y12 360 70 70 k=1 12 k =1 ϭ 417.94 Ϫ0.5 14 330 ∑ yk cos θk Ϫ20.78 Y10 12 Ϫ0.866 Y11 ∑ yk ϭ 212 cos 3θ V cos 3θ sin 3θ V sin 3θ 0 12 ∑ yk sin θk k =1 ϭ Ϫ278.53 12 ∑ yk cos 2θk k =1 ϭ Ϫ39 0 12 ∑ yk sin 2θk k =1 ϭ 29.43 12 12 ∑ yk cos 3θk ∑ yk sin 3θk k =1 k =1 ϭ Ϫ49 ϭ 55 Fourier Series 521 Substituting these values into the Fourier series: ϱ f( x ) ϭ a0 ϩ ∑ (an cos nx ϩ bn sin nx ) nϭ1 gives: v ‫ ؍‬17.67 ؉ 69.66 cos θ Ϫ 6.50 cos 2θ ؊ 8.17 cos 3θ ؉ … ؊46.42 sin θ ؉ 4.91 sin 2θ ؉ 9.17 sin 3θ ؉ (22) Note that in equation (22), (Ϫ46.42 sin θ ϩ 69.66 cos θ) comprises the fundamental, (4.91 sin 2θ Ϫ 6.50 cos 2θ) comprises the second harmonic and (9.17 sin 3θ Ϫ 8.17 cos 3θ) comprises the third harmonic It is shown in Chapter that: a sin ωt ϩ b cos ωt ϭ R sin(ωt ϩ α) where a ϭ R cos α, b ϭ R sin α, R ϭ a2 ϩ b2 and α ϭ tanϪ1 b a For the fundamental, R ϭ (Ϫ46.42)2 ϩ (69.66)2 ϭ 83.71 If a ϭ R cos α, then cos α ϭ a −46.42 ϭ which is negative, R 83.71 b 69.66 ϭ which is positive R 83.71 The only quadrant where cos α is negative and sin α is positive is the second quadrant b 69.66 ϭ 123.68Њ or 2.l6 rad Hence, ␣ ϭ tanϪ1 ϭ tanϪ1 Ϫ46.42 a and if b ϭ R sin α, then sin ␣ ϭ Thus, (Ϫ46.42 sin θ ϩ 69.66 cos θ) ϭ 83.71 sin(θ ϩ 2.16) By a similar method it may be shown that the second harmonic (4.91 sin 2θ Ϫ 6.50 cos 2θ) ϭ 8.15 sin(2θ Ϫ 0.92) harmonic and the third (9.17 sin 3θ Ϫ 8.17 cos 3θ) ϭ 12.28 sin(3θ Ϫ 0.73) Hence equation (22) may be re-written as: v ‫ ؍‬17.67 ؉ 83.71 sin( θ ؉ 2.16) ؉ 8.15 sin(2θ ؊ 0.92) ؉ 12.28 siin(3θ ؊ 0.73) volts which is the form normally used with complex waveforms 522 Engineering Mathematics Pocket Book 16.9 Complex waveform considerations It is sometimes possible to predict the harmonic content of a waveform on inspection of particular waveform characteristics If a periodic waveform is such that the area above the horizontal axis is equal to the area below then the mean value is zero Hence a0 ϭ (see Figure 16.14(a)) An even function is symmetrical about the vertical axis and contains no sine terms (see Figure 16.14(b)) An odd function is symmetrical about the origin and contains no cosine terms (see Figure 16.14(c)) f(x) ϭ f(x ϩ π) represents a waveform which repeats after half a cycle and only even harmonics are present (see Figure 16.14(d)) f(x) ϭ Ϫf(x ϩ π) represents a waveform for which the positive and negative cycles are identical in shape and only odd harmonics are present (see Figure 16.14(e)) f(x) f(x) π Ϫπ 2π x a0 ϭ (a) 2π x f(x) π 2π x Ϫ2πϪπ π f(x) π 2π x (d) Contains only even harmonics (c) Contains no cosine terms Ϫπ π (b) Contains no sine terms f(x) Ϫ2π Ϫπ 2π (e) Contains only odd harmonics x Figure 16.14 Fourier Series 523 Application: An alternating current i amperes is shown in Figure 16.15 Analyse the waveform into its constituent harmonics as far as and including the fifth harmonic, taking 30° intervals 10 i y5 y1 y2 y3 y4 Ϫ180 Ϫ120 Ϫ60 180 Ϫ150 Ϫ90 Ϫ30 30 60 90120150 Ϫ5 240 300 ␪° 210 270 y7 y8 y9 330 360 y11 y10 Ϫ10 Figure 16.15 With reference to Figure 16.15, the following characteristics are noted: (i) The mean value is zero since the area above the θ axis is equal to the area below it Thus the constant term, or d.c component, a0 ϭ (ii) Since the waveform is symmetrical about the origin the function i is odd, which means that there are no cosine terms present in the Fourier series (iii) The waveform is of the form f(θ) ϭ Ϫf(θ ϩ π) which means that only odd harmonics are present Investigating waveform characteristics has thus saved unnecessary calculations and in this case the Fourier series has only odd sine terms present, i.e i ϭ b1 sin θ ϩ b3 sin 3θ ϩ b5 sin 5θ ϩ A proforma, similar to Table 16.1, but without the ‘cosine terms’ columns and without the ‘even sine terms’ columns is shown in Table 16.2 up to, and including, the fifth harmonic, from which the Fourier coefficients b1, b3 and b5 can be determined Twelve co-ordinates are chosen and labelled y1, y2, y3, y12 as shown in Figure 16.15 524 Engineering Mathematics Pocket Book Table 16.2 θ i sin θ Y1 30 0.5 1 0.5 Y2 60 0.866 6.06 0 Ϫ0.866 Ϫ6.06 Y3 90 10 Ϫ1 Ϫ10 Y4 120 0.866 6.06 0 Ϫ0.866 Ϫ6.06 Y5 150 0.5 1 0.5 Y6 180 0 0 0 Ϫ1 Ϫ0.5 0 Ϫ10 0 Ordinate i sin θ 10 Y7 210 Ϫ2 Ϫ0.5 Y8 240 Ϫ7 Ϫ0.866 6.06 Y9 270 Ϫ10 Ϫ1 10 sin 3θ i sin 3θ sin 5θ i sin 5θ 10 0.866 Ϫ1 Ϫ6.06 10 Y10 300 Ϫ7 Ϫ0.866 6.06 Y11 330 Ϫ2 Ϫ0.5 Ϫ1 Ϫ0.5 Y12 360 0 0 0 12 ∑ y sin θ k k k =1 ϭ 48.24 12 ∑ y sin 3θ k k k =1 0.866 Ϫ6.06 12 ∑ y sin 5θ k k k =1 ϭ Ϫ12 ϭ Ϫ0.24 p From equation (21), bn ϭ Hence, b1 Ϸ b5 Ϸ ∑ ik sin nθk where p ϭ 12 p kϭ1 2 ( 48.24 ) ϭ 8.04 , b3 Ϸ (Ϫ12) ϭ Ϫ2.00 and 12 12 (Ϫ0.24 ) ϭ Ϫ0.04 12 Thus the Fourier series for current i is given by: i ‫ ؍‬8.04 sin θ ؊ 2.00 sin 3θ ؊ 0.04 sin 5θ Index Acceleration, 265 Acute angle, 105 Adjoint of matrix, 221 Algebraic method of successive approximations, 61 substitution, integration, 307 Alternate angles, 105 Amplitude, 129 And-function, 234 And-gate, 249 Angles of any magnitude, 125 elevation and depression, 113 Angle types, 105 Angstrom, Angular measure, velocity, 131 Applications of complex numbers, 211 Arc length of circle, 77, 79 Area, circle, 77 imperial, metric, of any triangle, 119 sector of circle, 77, 80 Areas of irregular figures, 96 plane figures, 73 similar shapes, 76 Areas under and between curves, 336 Argument, 209 Arithmetic progressions, 36 Astroid, 273 Astronomical constants, Asymptotes, 184 Average value of a waveform, 343 Bessel function, 402 Bessel’s equation, 402 Binary to decimal conversion, 65 hexadecimal conversion, 72 Binomial distribution, 434 series, 49 Bisection method, 59 Boolean algebra, 254 de Morgan’s laws, 241 Karnaugh maps, 242 laws and rules, 238 Cardioid, 177, 273 Cartesian and polar co-ordinates, 116 form of complex number, 206 Catenary, 37 Centroids, 350 Chain rule, 261 Change of limits, integration, 309 Changing products of sines and cosines into sums or differences, 146 sums or differences of sines and cosines into products, 147 Chi-square distribution, 454 Circle, arc length, 77 area of, 77 equation of, 81 sector of, 77 Circumference, 77 Coefficient of correlation, 443 Cofactor, 220 Combinational logic networks, 248 Combination of two periodic functions, 197 526 Index Complementary angles, 105 function, 379 Complex conjugate, 207 applications of, 211 equations, 206, 207 numbers, 206 Complex or exponential form of a Fourier series, 511 Complex waveform considerations, 522 Compound angle formulae, 39, 141 Compound angles, 141 Computer numbering systems, 65 Cone, 83, 85 Confidence levels, 448 Congruent triangles, 107 Constants, astronomical, mathematical, physical, Continued fractions, 24 Continuous function, 180 Contour map, 297 Convergents, 24 Conversions, Correlation, linear, 443 Corresponding angles, 105 Cosine rule, 119 waveform, 124, 128 Cramer’s rule, 230 Cubic equations, 170 Cuboid, 82 Cumulative frequency distribution, 421, 423 curve, 421 Cylinder, 82, 83 Cycloid, 273 Decile, 429 Decimal to binary conversion, 65 hexadecimal conversion, 71 via octal, 67 Definite integrals, 306 De Moivres theorem, 213 De Morgan’s laws, 241 Depression, angle of, 114 Derived units, Determinants, by, 2, 218 by, 3, 220 solution of simultaneous equations, 226 Determination of law, 152 involving logarithms, 153 Differential calculus, 258 function of a function, 261 products and quotients, 259 Differential equations, 366 d2y dy a ϩb ϩ cy ϭ 0, 375 dx dx dy dy a ϩb ϩ cy ϭ f( x ) , 379 dx dx dy ϭ f( x ), 366 dx dy ϭ f( y ), 367 dx dy ϭ f( x ).f( y ), 368 dx dy ϩ Py ϭ Q, 373 dx numerical methods, 385 dy P ϭ Q , 371 dx using Laplace transforms, 483 Differentiation, 258 of hyperbolic functions, 263 implicit functions, 276 inverse hyperbolic functions, 284 inverse trigonometric functions, 281, 282 logarithmic functions, 279 in parameters, 274 partial, 189 successive, 262 Direction cosines, 202 Discontinuous function, 180 Dividend, 20 Divisor, 20 Index Dot product, 200 Double angles, 39, 145 Elevation, angle of, 113 Ellipse, 73, 178, 273 Equation of a circle, 81 Equilateral triangle, 106 Euler’s method, 385 Euler-Cauchy method, 388 Evaluating trigonometric ratios, 110 Even function, 37, 180, 498 Exponential form of complex numbers, 215 Fourier series, 511 functions, 31 Extrapolation, 151 Factor theorem, 21 Final value theorem, 478 Finite discontinuities, 180 Fourier cosine series, 498 Fourier series for non-periodic function over period, 2π, 496 Fourier series for periodic function over period, 2π, 492 Fourier series for periodic function over period L, 504 Fourier sine series, 498 Fractional form of trigonometric ratios, 112 Frequency, 131 distribution, 422 polygon, 420, 422 Frobenius method, 398 Frustum of cone, 88 sphere, 92 Function of a function, 261 Gamma functions, 402 Gaussian elimination, 232 Geometric progressions, 47 Gradient of graph, 149 Graphical functions, 180 Graphs, cubic equations, 170 527 exponential functions, 31 hyperbolic functions, 37 logarithmic functions, 30 quadratic, 164, 166 simultaneous equations, 163 straight line, 149 trigonometric functions, 124 Greek alphabet, Grouped data, 420, 426 Half range Fourier series, 501, 508 Harmonic analysis, 518 Heat conduction equation, 411 Hectare, Heptagon, 74 Hexadecimal number, 69 to binary conversion, 72 decimal conversion, 70 Hexagon, 74, 75 Histogram, 420, 422, 428 Homogeneous first order differential equations, 371 Horizontal bar charts, 416, 417 Hyperbola, 179, 273 rectangular, 179, 273 Hyperbolic functions, 36 differentiation of, 263 identities, 38 solving equations, 39 Imaginary part of complex number, 206 Implicit functions, 276 Inflexion, point of, 267 Initial value theorem, 477 Integral calculus, 303 Integrals, algebraic substitutions, 307 definite, 306 by partial fractions, 317 by trigonometric and hyperbolic substitutions, 310 standard, 303 Integration by parts, 323 528 Index reduction formulae, 326 θ tan substitution, 319 Interior angles, 105 Interpolation, 150 Inverse functions, 182, 281 by, 3, 221 Laplace transforms, 480 of matrix, by, 2, 218 using partial fractions, 482 Inverse hyperbolic functions, 284 differentiation of, 285 Inverse trigonometric functions, 183 differentiation of, 281 Invert-gate, 249 Isosceles triangle, 106 Iterative methods, 58 Karnaugh maps, 242 Knot, Lagging angle, 129 Laplace’s equation, 413 Laplace transforms, 472 inverse, 480 Laws of growth and decay, 33 logarithms, 28, 279 Laws and rules of Boolean algebra, 239 Leading angle, 129 Least-squares regression line, 445 Legendre’s equation, 403 Legendre’s polynomials, 404 Leibniz-Maclaurin method, 395 Leibniz’s theorem, 394 Length, imperial, metric, L’Hopital’s rule, 57 Limiting values, 57 Linear correlation, 443 first order differential equations, 373 regression, 445 Litre, Logarithmic scales, 158 differentiation, 279 forms of inverse hyperbolic functions, 287 functions, 279 Logarithms, 28, 153 Logic circuits, 248 gates, 248 Maclaurin’s theorem, 54 Mann-Whitney test, 464 Mass, Mathematical constants, 5, 6, symbols, Matrices, 217 solution of simultaneous equations, 223 Maximum values, 102, 267, 295 Mean or average values, by integration, 343 Mean value, of a waveform, 101 statistics, 424, 426 Measures of central tendency, 424 Median, 424 Micron, Mid-ordinate rule, 95 numerical integration, 331 Minimum value, 267, 295 Minor of matrix, 220 Mode, 424 Modulus, 209 Nand-gate, 248, 253 Napierian logarithms, 32 Newton-Raphson method, 63 Nor-gate, 248, 253 Normal distribution, 437 Normals, 270 Nose-to-tail method, 189 Not-function, 234 Not-gate, 249 Numerical integration, 331 mid-ordinate rule, 331 Simpson’s rule, 332 Index trapezoidal rule, 331 using Maclaurin’s series, 56 Numerical methods for first order differential equations, 385 Numerical method of harmonic analysis, 518 Obtuse angle, 105 Octagon, 74, 75 Octal, 68 Octal to binary and decimal, 68 Odd function, 37, 181, 498 Ogive, 421, 424 Or-function, 234 Or-gate, 248 Pappus’s theorem, 354 Parabola, 273 Parametric equations, 273 Parallel axis theorem, 360 lines, 105 Parallelogram, 73 method for vector addition, 189 Partial differential equations, 405 differentiation, 289 fractions, 41, 317, 482 Particular integral, 379 solution of differential equation, 375 Pentagon, 74 Percentage component bar chart, 416, 418 Percentile, 429 Period, 129, 180 Periodic functions, 180 Perpendicular axis theorem, 360 Pictograms, 416 Pie diagram, 416, 420 Point of inflexion, 267 Poisson distribution, 435 Polar co-ordinates, 115 curves, 171 form, 209 Polygons, 73 529 Polynomial division, 20 Power series for ex, 31 methods of solving differential equations, 394 Prefixes, Prismoidal rule, 99 Probability, 431 Product-moment formula, 443 Products and quotients, 259 Pyramid, 82, 84 Pythagoras’s theorem, 108 Quadratic equations, 25, 166 graphs, 164 Quadrilateral, 73 Quartiles, 429 Quotients, 24 Radian measure, 77, 78 Rates of change using differentiation, 264 partial differentiation, 293 Real part of complex number, 206 Reciprocal ratios, 110 Rectangle, 73 Rectangular form of complex numbers, 206 hyperbola, 179, 273 prism, 82, 83 Recurrence relation, 396 Reduction formulae, 326 Reflex angle, 105 Regression, linear, 445 Relationship between trigonometric and hyperbolic functions, 139 Relative velocity, 195 Remainder theorem, 23 Resolution of vectors, 191 Right angle, 105 Right-angled triangle solution, 113 Rodrigue’s formula, 404 Root mean square values, 345 Runge-Kutta method, 390 530 Index Saddle points, 296 Sampling and estimation theories, 447 Scalar product of two vectors, 200 quantities, 188 Scalene triangle, 106 Second moments of area, 359 Sector of circle, 77 Semi-interquartile range, 430 Sign test, 457 Similar triangles, 107 Simpson’s rule, 96, 98 numerical integration, 332 Simultaneous differential equations by Laplace transforms, 487 Simultaneous equations, by Cramer’s rule, 230 determinants, 226 Gaussian elimination, 232 graphical solution, 163 matrices, 223 Sine rule, 119 waveform, 102, 124, 127 Sinusoidal form A sin (ωt Ϯ φ), 131 SI units, Small changes using differentiation, 272 partial differentiation, 294 Solution of right-angled triangles, 113 Solving equations by algebraic method, 61 bisection method, 59 containing hyperbolic functions, 39 iterative methods, 58 Newton-Raphson method, 63 quadratics, 25 Speed, Sphere, 83 Standard derivatives, 258 deviation, 424, 426 error of the means, 447 integrals, 303 Laplace transforms, 472 Stationary points, 267 Straight line graphs, 149 Student’s t distribution, 451 Successive differentiation, 262 Supplementary angles, 105 Surd form, 112 Symbols, acoustics, 14 atomic and nuclear physics, 17 electricity and magnetism, 13 light related electromagnetic radiations, 14 mathematical, mechanics, 11 molecular physics, 16 nuclear reactions and ionising radiations, 19 periodic and related phenomena, 11 physical chemistry, 15 quantities, 10 space and time, 10 thermodynamics, 12 Tally diagram, 421 Tangents, 270 Tangent waveform, 124 θ Tan substitution, 319 Theorem of Pappus, 354 Pythagoras, 108 Total differential, 292 Transpose of matrix, 221 Transversal, 105 Trapezium, 73 Trapezoidal rule, 95, 518 numerical integration, 331 Triangle, 73 Triangles, properties of, 106 Trigonometric and hyperbolic substitutions, integration, 310 Trigonometric ratios, 109 equations, 134 evaluating, 110 fractional and surd form of, 112 identities, 38, 134 Index Truth tables, 234 Turning points, 267 Ungrouped data, 416 Unit matrix, 219 Universal logic gate, 253 Vector addition, 189 products, 203 subtraction, 192 Vectors, 188 Velocity and acceleration, using integration, 265, Vertical bar chart, 416, 418 Vertically opposite angles, 105 531 Volume, Volumes and surface areas of frusta of pyramids and cones, 88 irregular solids, 98 using Simpson’s rule, 96 Volumes of similar shapes, 87 solids of revolution, 347 Wave equation, 406 Wilcoxon signed-rank test, 460 Xor-gate, 250 Xnor-gate, 250 Zone of a sphere, 92 .. .Engineering Mathematics Pocket Book Fourth Edition John Bird This page intentionally left blank Engineering Mathematics Pocket Book Fourth edition John Bird BSc(Hons),... 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. .. Preface Engineering Mathematics Pocket Book 4th Edition is intended to provide students, technicians, scientists and engineers with a readily available reference to the essential engineering mathematics