Higher Engineering Mathematics - BS Grewal

Higher Engineering Mathematics

In memory of Elizabeth

Higher Engineering Mathematics Sixth Edition John Bird, BSc (Hons), CMath, CEng, CSci, 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 The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 2010 Copyright © 2010, 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. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. 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 catalogue record for this book is available from the Library of Congress. ISBN: 978-1-85-617767-2 For information on all Newnes publications visit our Web site at www.elsevierdirect.com Typeset by: diacriTech, India Printed and bound in China 10 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1

Contents Preface xiii 6 Arithmetic and geometric progressions 51 6.1 Arithmetic progressions 51 Syllabus guidance xv 6.2 Worked problems on arithmetic progressions 51 1 Algebra 1 6.3 Further worked problems on arithmetic 1.1 Introduction 1 progressions 52 1.2 Revision of basic laws 1 6.4 Geometric progressions 54 1.3 Revision of equations 3 6.5 Worked problems on geometric 1.4 Polynomial division 6 progressions 55 1.5 The factor theorem 8 6.6 Further worked problems on geometric 1.6 The remainder theorem 10 progressions 56 2 Partial fractions 13 7 The binomial series 58 2.1 Introduction to partial fractions 13 7.1 Pascal’s triangle 58 2.2 Worked problems on partial fractions with 7.2 The binomial series 59 linear factors 13 7.3 Worked problems on the binomial series 59 2.3 Worked problems on partial fractions with 7.4 Further worked problems on the binomial repeated linear factors 16 series 62 2.4 Worked problems on partial fractions with 7.5 Practical problems involving the binomial quadratic factors 17 theorem 64 3 Logarithms 20 Revision Test 2 67 3.1 Introduction to logarithms 20 3.2 Laws of logarithms 22 8 Maclaurin’s series 68 3.3 Indicial equations 24 8.1 Introduction 68 3.4 Graphs of logarithmic functions 25 8.2 Derivation of Maclaurin’s theorem 68 8.3 Conditions of Maclaurin’s series 69 4 Exponential functions 27 8.4 Worked problems on Maclaurin’s series 69 4.1 Introduction to exponential functions 27 8.5 Numerical integration using Maclaurin’s 4.2 The power series for ex 28 series 73 4.3 Graphs of exponential functions 29 8.6 Limiting values 74 4.4 Napierian logarithms 31 4.5 Laws of growth and decay 34 9 Solving equations by iterative methods 77 4.6 Reduction of exponential laws to 9.1 Introduction to iterative methods 77 linear form 37 9.2 The bisection method 77 9.3 An algebraic method of successive Revision Test 1 40 approximations 81 9.4 The Newton-Raphson method 84 5 Hyperbolic functions 41 5.1 Introduction to hyperbolic functions 41 10 Binary, octal and hexadecimal 87 5.2 Graphs of hyperbolic functions 43 10.1 Introduction 87 5.3 Hyperbolic identities 45 10.2 Binary numbers 87 5.4 Solving equations involving hyperbolic 10.3 Octal numbers 90 functions 47 10.4 Hexadecimal numbers 92 5.5 Series expansions for cosh x and sinh x 49 Revision Test 3 96

vi Contents 11 Introduction to trigonometry 97 15.5 Worked problems (ii) on trigonometric 156 11.1 Trigonometry 97 equations 157 11.2 The theorem of Pythagoras 97 157 11.3 Trigonometric ratios of acute angles 98 15.6 Worked problems (iii) on trigonometric 11.4 Evaluating trigonometric ratios 100 equations 11.5 Solution of right-angled triangles 105 11.6 Angles of elevation and depression 106 15.7 Worked problems (iv) on trigonometric 11.7 Sine and cosine rules 108 equations 11.8 Area of any triangle 108 11.9 Worked problems on the solution of 16 The relationship between trigonometric and 159 triangles and finding their areas 109 hyperbolic functions 11.10 Further worked problems on solving 16.1 The relationship between trigonometric 159 triangles and finding their areas 110 and hyperbolic functions 160 11.11 Practical situations involving trigonometry 111 16.2 Hyperbolic identities 11.12 Further practical situations involving trigonometry 113 17 Compound angles 163 17.1 Compound angle formulae 163 12 Cartesian and polar co-ordinates 117 17.2 Conversion of a sinωt + b cosωt into 12.1 Introduction 117 R sin(ωt + α) 165 12.2 Changing from Cartesian into polar 17.3 Double angles 169 co-ordinates 117 17.4 Changing products of sines and cosines 12.3 Changing from polar into Cartesian into sums or differences 170 co-ordinates 119 12.4 Use of Pol/Rec functions on calculators 120 17.5 Changing sums or differences of sines and 171 cosines into products 173 17.6 Power waveforms in a.c. circuits Revision Test 5 177 13 The circle and its properties 122 18 Functions and their curves 178 13.1 Introduction 122 18.1 Standard curves 178 13.2 Properties of circles 122 18.2 Simple transformations 181 13.3 Radians and degrees 123 18.3 Periodic functions 186 13.4 Arc length and area of circles and sectors 124 18.4 Continuous and discontinuous functions 186 13.5 The equation of a circle 127 18.5 Even and odd functions 186 13.6 Linear and angular velocity 129 18.6 Inverse functions 188 13.7 Centripetal force 130 18.7 Asymptotes 190 18.8 Brief guide to curve sketching 196 Revision Test 4 133 18.9 Worked problems on curve sketching 197 14 Trigonometric waveforms 134 19 Irregular areas, volumes and mean values of 203 14.1 Graphs of trigonometric functions 134 waveforms 203 14.2 Angles of any magnitude 135 19.1 Areas of irregular figures 205 14.3 The production of a sine and cosine wave 137 19.2 Volumes of irregular solids 206 14.4 Sine and cosine curves 138 19.3 The mean or average value of a waveform 14.5 Sinusoidal form A sin(ωt ± α) 143 14.6 Harmonic synthesis with complex Revision Test 6 212 waveforms 146 15 Trigonometric identities and equations 152 20 Complex numbers 213 15.1 Trigonometric identities 152 20.1 Cartesian complex numbers 213 15.2 Worked problems on trigonometric 20.2 The Argand diagram 214 identities 152 20.3 Addition and subtraction of complex 15.3 Trigonometric equations 154 numbers 214 15.4 Worked problems (i) on trigonometric 20.4 Multiplication and division of complex equations 154 numbers 216

Contents vii 20.5 Complex equations 217 25.4 Determining resultant phasors by the sine 20.6 The polar form of a complex number 218 20.7 Multiplication and division in polar form 220 and cosine rules 268 20.8 Applications of complex numbers 221 25.5 Determining resultant phasors by horizontal and vertical components 270 25.6 Determining resultant phasors by complex 21 De Moivre’s theorem 225 numbers 272 21.1 Introduction 225 21.2 Powers of complex numbers 225 26 Scalar and vector products 275 21.3 Roots of complex numbers 226 26.1 The unit triad 275 21.4 The exponential form of a complex 26.2 The scalar product of two vectors 276 number 228 26.3 Vector products 280 26.4 Vector equation of a line 283 22 The theory of matrices and determinants 231 Revision Test 8 286 22.1 Matrix notation 231 22.2 Addition, subtraction and multiplication of matrices 231 27 Methods of differentiation 287 27.1 Introduction to calculus 287 22.3 The unit matrix 235 27.2 The gradient of a curve 287 27.3 Differentiation from first principles 288 22.4 The determinant of a 2 by 2 matrix 235 27.4 Differentiation of common functions 289 27.5 Differentiation of a product 292 22.5 The inverse or reciprocal of a 2 by 2 matrix 236 27.6 Differentiation of a quotient 293 27.7 Function of a function 295 22.6 The determinant of a 3 by 3 matrix 237 27.8 Successive differentiation 296 22.7 The inverse or reciprocal of a 3 by 3 matrix 239 23 The solution of simultaneous equations by 241 28 Some applications of differentiation 299 matrices and determinants 241 28.1 Rates of change 299 23.1 Solution of simultaneous equations by 243 28.2 Velocity and acceleration 300 matrices 247 28.3 Turning points 303 248 28.4 Practical problems involving maximum 23.2 Solution of simultaneous equations by and minimum values 307 determinants 28.5 Tangents and normals 311 28.6 Small changes 312 23.3 Solution of simultaneous equations using Cramers rule 23.4 Solution of simultaneous equations using the Gaussian elimination method Revision Test 7 250 24 Vectors 251 29 Differentiation of parametric equations 315 24.1 Introduction 251 29.1 Introduction to parametric equations 315 24.2 Scalars and vectors 251 29.2 Some common parametric equations 315 24.3 Drawing a vector 251 29.3 Differentiation in parameters 315 24.4 Addition of vectors by drawing 252 29.4 Further worked problems on 24.5 Resolving vectors into horizontal and differentiation of parametric equations 318 vertical components 254 24.6 Addition of vectors by calculation 255 30 Differentiation of implicit functions 320 24.7 Vector subtraction 260 30.1 Implicit functions 320 24.8 Relative velocity 262 30.2 Differentiating implicit functions 320 24.9 i, j and k notation 263 30.3 Differentiating implicit functions containing products and quotients 321 30.4 Further implicit differentiation 322 25 Methods of adding alternating waveforms 265 31 Logarithmic differentiation 325 25.1 Combination of two periodic functions 265 31.1 Introduction to logarithmic differentiation 325 25.2 Plotting periodic functions 265 31.2 Laws of logarithms 325 25.3 Determining resultant phasors by drawing 267 31.3 Differentiation of logarithmic functions 325

viii Contents 31.4 Differentiation of further logarithmic 326 38 Some applications of integration 375 functions 328 38.1 Introduction 375 38.2 Areas under and between curves 375 31.5 Differentiation of [ f (x)]x 38.3 Mean and r.m.s. values 377 38.4 Volumes of solids of revolution 378 Revision Test 9 330 38.5 Centroids 380 38.6 Theorem of Pappus 381 32 Differentiation of hyperbolic functions 331 38.7 Second moments of area of regular 32.1 Standard differential coefficients of 331 sections 383 hyperbolic functions 332 39 Integration using algebraic substitutions 392 32.2 Further worked problems on 39.1 Introduction 392 differentiation of hyperbolic functions 39.2 Algebraic substitutions 392 39.3 Worked problems on integration using 33 Differentiation of inverse trigonometric and 334 algebraic substitutions 392 hyperbolic functions 334 39.4 Further worked problems on integration 33.1 Inverse functions using algebraic substitutions 394 334 39.5 Change of limits 395 33.2 Differentiation of inverse trigonometric functions 339 Revision Test 11 397 33.3 Logarithmic forms of the inverse 341 hyperbolic functions 33.4 Differentiation of inverse hyperbolic functions 34 Partial differentiation 345 40 Integration using trigonometric and hyperbolic 398 34.1 Introduction to partial derivatives 345 substitutions 398 34.2 First order partial derivatives 345 40.1 Introduction 398 34.3 Second order partial derivatives 348 40.2 Worked problems on integration of sin2 x, 400 cos2 x, tan2 x and cot2 x 401 35 Total differential, rates of change and small 351 402 changes 351 40.3 Worked problems on powers of sines and 404 35.1 Total differential 352 cosines 404 35.2 Rates of change 354 406 35.3 Small changes 40.4 Worked problems on integration of products of sines and cosines 36 Maxima, minima and saddle points for functions 357 of two variables 357 40.5 Worked problems on integration using the 36.1 Functions of two independent variables 358 sin θ substitution 36.2 Maxima, minima and saddle points 359 40.6 Worked problems on integration using tan θ substitution 36.3 Procedure to determine maxima, minima 359 and saddle points for functions of two 40.7 Worked problems on integration using the variables 361 sinh θ substitution 36.4 Worked problems on maxima, minima 40.8 Worked problems on integration using the and saddle points for functions of two cosh θ substitution variables 41 Integration using partial fractions 409 36.5 Further worked problems on maxima, 41.1 Introduction 409 minima and saddle points for functions of 41.2 Worked problems on integration using 409 two variables partial fractions with linear factors 41.3 Worked problems on integration using 411 Revision Test 10 367 partial fractions with repeated linear 412 factors 414 37 Standard integration 368 41.4 Worked problems on integration using 414 37.1 The process of integration 368 partial fractions with quadratic factors 415 37.2 The general solution of integrals of the 368 42 The t = tan θ substitution form axn 369 2 372 37.3 Standard integrals 42.1 Introduction 37.4 Definite integrals 42.2 Worked problems on the t = tan θ 2 substitution

Contents ix 42.3 Further worked problems on the t = tan θ 48 Linear first order differential equations 456 2 48.1 Introduction 456 substitution 416 48.2 Procedure to solve differential equations 457 Revision Test 12 419 of the form dy + P y = Q dx 457 43 Integration by parts 420 458 43.1 Introduction 420 48.3 Worked problems on linear first order 43.2 Worked problems on integration by parts 420 differential equations 43.3 Further worked problems on integration by parts 422 48.4 Further worked problems on linear first order differential equations 44 Reduction formulae 426 49 Numerical methods for first order differential 461 44.1 Introduction 426 equations 461 49.1 Introduction 461 44.2 Using reduction formulae for integrals of 426 49.2 Euler’s method 462 the form xnex dx 49.3 Worked problems on Euler’s method 466 427 49.4 An improved Euler method 471 44.3 Using reduction formulae for integrals of 49.5 The Runge-Kutta method the form xncos x dx and xn sin x dx 429 432 Revision Test 14 476 44.4 Using reduction formulae for integrals of the form sinn x dx and cosn x dx 50 Second order differential equations of the form d2y dy 44.5 Further reduction formulae a dx2 +b dx + cy=0 477 45 Numerical integration 435 50.1 Introduction 477 45.1 Introduction 435 45.2 The trapezoidal rule 435 50.2 Procedure to solve differential equations 45.3 The mid-ordinate rule 437 45.4 Simpson’s rule 439 of the form a d2y + b dy + cy = 0 478 dx2 dx 50.3 Worked problems on differential equations Revision Test 13 443 of the form a d2y + b dy + cy = 0 478 dx2 dx 50.4 Further worked problems on practical differential equations of the form 46 Solution of first order differential equations by 444 a d2 y + b dy + cy = 0 480 444 dx2 dx separation of variables 445 51 Second order differential equations of the form 483 46.1 Family of curves 445 d2y dy 46.2 Differential equations 447 a dx2 + b dx + cy= f (x) 46.3 The solution of equations of the form 449 51.1 Complementary function and particular 483 dy = f (x) integral dx 51.2 Procedure to solve differential equations 46.4 The solution of equations of the form dy d2y dy 483 = f (y) of the form a dx2 + b dx + cy = f (x) dx 51.3 Worked problems on differential equations 46.5 The solution of equations of the form dy = f (x) · f (y) d2y dy dx dx2 dx of the form a + b + cy = f (x) 47 Homogeneous first order differential equations 452 where f (x) is a constant or polynomial 484 47.1 Introduction 452 51.4 Worked problems on differential equations 47.2 Procedure to solve differential equations of the form a d2y + b dy + cy = f (x) of the form P dy = Q dx2 dx dx 452 where f (x) is an exponential function 486 47.3 Worked problems on homogeneous first 51.5 Worked problems on differential equations order differential equations 452 d2y dy dx2 dx 47.4 Further worked problems on homogeneous of the form a + b + cy = f (x) first order differential equations 454 where f (x) is a sine or cosine function 488

x Contents 51.6 Worked problems on differential equations 57 The binomial and Poisson distributions 556 57.1 The binomial distribution 556 of the form a d2 y +b dy + cy = f (x) 57.2 The Poisson distribution 559 dx2 dx where f (x) is a sum or a product 490 58 The normal distribution 562 58.1 Introduction to the normal distribution 562 52 Power series methods of solving ordinary 493 58.2 Testing for a normal distribution 566 differential equations 493 52.1 Introduction 59 Linear correlation 570 52.2 Higher order differential coefficients as 493 59.1 Introduction to linear correlation 570 series 495 59.2 The product-moment formula for 52.3 Leibniz’s theorem determining the linear correlation 570 52.4 Power series solution by the 497 coefficient Leibniz–Maclaurin method 59.3 The significance of a coefficient of 571 52.5 Power series solution by the Frobenius 500 correlation 571 method 506 59.4 Worked problems on linear correlation 52.6 Bessel’s equation and Bessel’s functions 575 52.7 Legendre’s equation and Legendre 511 60 Linear regression 575 polynomials 60.1 Introduction to linear regression 575 60.2 The least-squares regression lines 576 53 An introduction to partial differential equations 515 60.3 Worked problems on linear regression 53.1 Introduction 515 53.2 Partial integration 515 Revision Test 17 581 53.3 Solution of partial differential equations by direct partial integration 516 61 Introduction to Laplace transforms 582 53.4 Some important engineering partial 61.1 Introduction 582 differential equations 518 61.2 Definition of a Laplace transform 582 53.5 Separating the variables 518 61.3 Linearity property of the Laplace 53.6 The wave equation 519 transform 582 53.7 The heat conduction equation 523 61.4 Laplace transforms of elementary 53.8 Laplace’s equation 525 functions 582 61.5 Worked problems on standard Laplace Revision Test 15 528 transforms 583 54 Presentation of statistical data 529 62 Properties of Laplace transforms 587 54.1 Some statistical terminology 529 62.1 The Laplace transform of eat f (t) 587 54.2 Presentation of ungrouped data 530 62.2 Laplace transforms of the form eat f (t) 587 54.3 Presentation of grouped data 534 589 62.3 The Laplace transforms of derivatives 591 62.4 The initial and final value theorems 55 Measures of central tendency and dispersion 541 63 Inverse Laplace transforms 593 55.1 Measures of central tendency 541 55.2 Mean, median and mode for discrete data 541 63.1 Definition of the inverse Laplace transform 593 55.3 Mean, median and mode for grouped data 542 55.4 Standard deviation 544 63.2 Inverse Laplace transforms of simple 55.5 Quartiles, deciles and percentiles 546 functions 593 63.3 Inverse Laplace transforms using partial fractions 596 56 Probability 548 63.4 Poles and zeros 598 56.1 Introduction to probability 548 56.2 Laws of probability 549 64 The solution of differential equations using 600 56.3 Worked problems on probability 549 Laplace transforms 600 56.4 Further worked problems on probability 551 64.1 Introduction 600 Revision Test 16 554 64.2 Procedure to solve differential equations by using Laplace transforms 600 64.3 Worked problems on solving differential equations using Laplace transforms

Contents xi 65 The solution of simultaneous differential 68.2 Fourier cosine and Fourier sine series 623 68.3 Half-range Fourier series 626 equations using Laplace transforms 605 65.1 Introduction 605 65.2 Procedure to solve simultaneous 69 Fourier series over any range 630 69.1 Expansion of a periodic function of 630 differential equations using Laplace period L 634 transforms 605 69.2 Half-range Fourier series for functions defined over range L 65.3 Worked problems on solving simultaneous differential equations by using Laplace transforms 605 70 A numerical method of harmonic analysis 637 Revision Test 18 610 70.1 Introduction 637 70.2 Harmonic analysis on data given in tabular or graphical form 637 66 Fourier series for periodic functions of 611 70.3 Complex waveform considerations 641 period 2π 611 66.1 Introduction 611 71 The complex or exponential form of a 644 611 Fourier series 644 66.2 Periodic functions 71.1 Introduction 644 612 71.2 Exponential or complex notation 645 66.3 Fourier series 71.3 The complex coefficients 649 71.4 Symmetry relationships 652 66.4 Worked problems on Fourier series of 71.5 The frequency spectrum 653 periodic functions of period 2π 71.6 Phasors 67 Fourier series for a non-periodic function over 617 Revision Test 19 658 range 2π 617 67.1 Expansion of non-periodic functions 617 67.2 Worked problems on Fourier series of non-periodic functions over a range of 2π 68 Even and odd functions and half-range 623 Essential formulae 659 Fourier series 623 Index 675 68.1 Even and odd functions

xii Contents Website Chapters 72 Inequalities 1 74.3 The sampling distribution of the means 29 72.1 Introduction to inequalities 1 74.4 The estimation of population parameters 33 72.2 Simple inequalities 1 38 72.3 Inequalities involving a modulus 2 based on a large sample size 72.4 Inequalities involving quotients 3 74.5 Estimating the mean of a population based 72.5 Inequalities involving square functions 4 72.6 Quadratic inequalities 5 on a small sample size 73 Boolean algebra and logic circuits 7 75 Significance testing 42 73.1 Boolean algebra and switching circuits 7 75.1 Hypotheses 42 73.2 Simplifying Boolean expressions 12 75.2 Type I and Type II errors 42 73.3 Laws and rules of Boolean algebra 12 75.3 Significance tests for population means 49 73.4 De Morgan’s laws 14 75.4 Comparing two sample means 54 73.5 Karnaugh maps 15 73.6 Logic circuits 19 76 Chi-square and distribution-free tests 59 73.7 Universal logic gates 23 76.1 Chi-square values 59 76.2 Fitting data to theoretical distributions 60 Revision Test 20 28 76.3 Introduction to distribution-free tests 67 76.4 The sign test 68 76.5 Wilcoxon signed-rank test 71 76.6 The Mann-Whitney test 75 74 Sampling and estimation theories 29 Revision Test 21 82 74.1 Introduction 29 74.2 Sampling distributions 29

Preface This sixth edition of ‘Higher Engineering Mathe- Each topic considered in the text is presented in a way matics’ covers essential mathematical material suitable that assumes in the reader only knowledge attained in for students studying Degrees, Foundation Degrees, BTEC National Certificate/Diploma, or similar, in an Higher National Certificate and Diploma courses in Engineering discipline. Engineering disciplines. ‘Higher Engineering Mathematics 6th Edition’ pro- In this edition the material has been ordered into the vides a follow-up to ‘Engineering Mathematics 6th following twelve convenient categories: number and Edition’. algebra, geometry and trigonometry, graphs, complex numbers, matrices and determinants, vector geometry, This textbook contains some 900 worked prob- differential calculus, integral calculus, differential equa- lems, followed by over 1760 further problems (with tions, statistics and probability, Laplace transforms and answers), arranged within 238 Exercises. Some 432 Fourier series. New material has been added on log- line diagrams further enhance understanding. arithms and exponential functions, binary, octal and hexadecimal, vectors and methods of adding alternat- A sample of worked solutions to over 1100 of the fur- ing waveforms. Another feature is that a free Internet ther problems has been prepared and can be accessed download is available of a sample (over 1100) of the free via the Internet (see next page). further problems contained in the book. At the end of the text, a list of Essential Formulae is The primary aim of the material in this text is to included for convenience of reference. provide the fundamental analytical and underpinning knowledge and techniques needed to successfully com- At intervals throughout the text are some 19 Revision plete scientific and engineering principles modules of Tests (plus two more in the website chapters) to check Degree, Foundation Degree and Higher National Engi- understanding. For example, Revision Test 1 covers neering programmes. The material has been designed the material in Chapters 1 to 4, Revision Test 2 cov- to enable students to use techniques learned for the ers the material in Chapters 5 to 7, Revision Test 3 analysis, modelling and solution of realistic engineering covers the material in Chapters 8 to 10, and so on. An problems at Degree and Higher National level. It also Instructor’s Manual, containing full solutions to the aims to provide some of the more advanced knowledge Revision Tests, is available free to lecturers adopting required for those wishing to pursue careers in mechan- this text (see next page). ical engineering, aeronautical engineering, electronics, communications engineering, systems engineering and Due to restriction of extent, five chapters that appeared all variants of control engineering. in the fifth edition have been removed from the text and placed on the website. For chapters on Inequali- In Higher Engineering Mathematics 6th Edition, the- ties, Boolean algebra and logic circuits, Sampling and ory is introduced in each chapter by a full outline of estimation theories, Significance testing and Chi-square essential definitions, formulae, laws, procedures etc. and distribution-free tests (see next page). The theory is kept to a minimum, for problem solving is extensively used to establish and exemplify the theory. ‘Learning by example’ is at the heart of ‘Higher It is intended that readers will gain real understand- Engineering Mathematics 6th Edition’. ing through seeing problems solved and then through solving similar problems themselves. JOHN BIRD Royal Naval School of Marine Engineering, Access to software packages such as Maple, Mathemat- ica and Derive, or a graphics calculator, will enhance HMS Sultan, understanding of some of the topics in this text. formerly University of Portsmouth and Highbury College, Portsmouth

xiv Preface Sample of worked Solutions to Exercises Free web downloads Within the text (plus the website chapters) are Extra material available on the Internet at: some 1900 further problems arranged within www.booksite.elsevier.com/newnes/bird. 260 Exercises. A sample of over 1100 worked solutions has been prepared and can be accessed It is recognised that the level of understanding free via the Internet. To access these worked of algebra on entry to higher courses is often solutions visit the website. inadequate. Since algebra provides the basis of so much of higher engineering studies, it is a situation Instructor’s manual that often needs urgent attention. Lack of space has prevented the inclusion of more basic algebra This provides fully worked solutions and mark topics in this textbook; it is for this reason that scheme for all the Revision Tests in this book some algebra topics – solution of simple, simul- (plus 2 from the website chapters), together with taneous and quadratic equations and transposition solutions to the Remedial Algebra Revision Test of formulae – have been made available to all via mentioned above. The material is available to lec- the Internet. Also included is a Remedial Algebra turers only. To obtain a password please visit the Revision Test to test understanding. To access the website with the following details: course title, Algebra material visit the website. number of students, your job title and work postal address. Five extra chapters To download the Instructor’s Manual visit the Chapters on Inequalities, Boolean Algebra and website and enter the book title in the search box. logic circuits, Sampling and Estimation theo- ries, Significance testing, and Chi-square and distribution-free tests are available to download at the website.

Syllabus Guidance This textbook is written for undergraduate engineering degree and foundation degree courses; however, it is also most appropriate for HNC/D studies and three syllabuses are covered. The appropriate chapters for these three syllabuses are shown in the table below. Chapter Analytical Further Engineering Methods Analytical Mathematics 1. Algebra for Methods for 2. Partial fractions Engineers Engineers (Continued ) 3. Logarithms 4. Exponential functions × × 5. Hyperbolic functions × × 6. Arithmetic and geometric progressions × 7. The binomial series × × 8. Maclaurin’s series × × 9. Solving equations by iterative methods × × 10. Binary, octal and hexadecimal × × 11. Introduction to trigonometry × × 12. Cartesian and polar co-ordinates × 13. The circle and its properties × × 14. Trigonometric waveforms × × 15. Trigonometric identities and equations × 16. The relationship between trigonometric and hyperbolic × × functions × 17. Compound angles 18. Functions and their curves × 19. Irregular areas, volumes and mean values of waveforms 20. Complex numbers 21. De Moivre’s theorem 22. The theory of matrices and determinants 23. The solution of simultaneous equations by matrices and determinants 24. Vectors 25. Methods of adding alternating waveforms

xvi Syllabus Guidance Chapter Analytical Further Engineering Methods Analytical Mathematics 26. Scalar and vector products for Methods for Engineers Engineers × 27. Methods of differentiation × × × × 28. Some applications of differentiation × × × × 29. Differentiation of parametric equations × × × × × 30. Differentiation of implicit functions × × × × (Continued ) 31. Logarithmic differentiation × × × 32. Differentiation of hyperbolic functions × × 33. Differentiation of inverse trigonometric and hyperbolic × functions × × 34. Partial differentiation × 35. Total differential, rates of change and small changes 36. Maxima, minima and saddle points for functions of two variables 37. Standard integration 38. Some applications of integration 39. Integration using algebraic substitutions 40. Integration using trigonometric and hyperbolic substitutions 41. Integration using partial fractions 42. The t = tan θ/2 substitution 43. Integration by parts 44. Reduction formulae 45. Numerical integration 46. Solution of first order differential equations by separation of variables 47. Homogeneous first order differential equations 48. Linear first order differential equations 49. Numerical methods for first order differential equations 50. Second order differential equations of the form d2y dy a dx2 + b dx + cy = 0 51. Second order differential equations of the form d2y + b dy + cy = f (x) a dx2 dx 52. Power series methods of solving ordinary differential equations 53. An introduction to partial differential equations 54. Presentation of statistical data

Syllabus Guidance xvii Chapter Analytical Further Engineering Methods Analytical Mathematics 55. Measures of central tendency and dispersion for Methods for 56. Probability Engineers Engineers × 57. The binomial and Poisson distributions × 58. The normal distribution × × × 59. Linear correlation × × 60. Linear regression × × 61. Introduction to Laplace transforms × × 62. Properties of Laplace transforms × × 63. Inverse Laplace transforms × × 64. Solution of differential equations using Laplace transforms × 65. The solution of simultaneous differential equations using × × × × Laplace transforms × 66. Fourier series for periodic functions of period 2π 67. Fourier series for non-periodic functions over range 2π 68. Even and odd functions and half-range Fourier series 69. Fourier series over any range 70. A numerical method of harmonic analysis 71. The complex or exponential form of a Fourier series Website Chapters 72. Inequalities 73. Boolean algebra and logic circuits 74. Sampling and estimation theories 75. Significance testing 76. Chi-square and distribution-free tests

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Chapter 1 Algebra 1.1 Introduction 3x + 2y x−y In this chapter, polynomial division and the factor and remainder theorems are explained (in Sections 1.4 Multiply by x → 3x2 + 2x y to 1.6). However, before this, some essential algebra revision on basic laws and equations is included. Multiply by −y → −3x y − 2y2 For further Algebra revision, go to website: Adding gives: 3x2 − xy − 2y2 http://books.elsevier.com/companions/0750681527 Alternatively, 1.2 Revision of basic laws (3x + 2y)(x − y) = 3x2 − 3x y + 2x y − 2y2 = 3x2 − xy − 2y2 (a) Basic operations and laws of indices The laws of indices are: a3b2c4 (i) am × an = am+n (ii) am = am−n Problem 3. Simplify abc−2 and evaluate when an 1 a = 3, b = 8 and c = 2. √ (iii) (am)n = am×n (iv) m = n am an (v) a−n = 1 (vi) a0 = 1 a3b2c4 = a3−1b2−1c4−(−2) = a2bc6 an abc−2 When a = 3, b = 1 and c = 2, 8 Problem 1. Evaluate 4a2bc3−2ac when a = 2, a2bc6 = (3)2 1 (2)6 = (9) 1 (64) = 72 8 8 b= 1 and c = 1 1 2 2 4a2bc3 − 2ac = 4(2)2 1 3 3 3 x2y3 + x y2 Problem 4. Simplify 22 − 2(2) 2 xy = 4 × 2 × 2 × 3 × 3 × 3 12 x2y3 + x y2 x2y3 x y2 − =+ 2×2×2×2 2 xy xy xy = 27 − 6 = 21 = x2−1 y3−1 + x1−1 y2−1 = xy2 + y or y(xy + 1) Problem 2. Multiply 3x + 2y by x − y.

2 Higher Engineering Mathematics (x 2 √ √ x 3 y2) (b) Brackets, factorization and precedence y)( Problem 5. Simplify Problem 6. Simplify a2 − (2a − ab) − a(3b + a). 5 y3) 1 (x 2 a2 − (2a − ab) − a(3b + a) = a2 − 2a + ab − 3ab − a2 (x2√y)(√x 3 y2) x2 y 1 x 1 y 2 = −2a − 2ab or −2a(1 + b) 2 2 3 = 53 Problem 7. Remove the brackets and simplify the (x 5 y 3 ) 1 x2 y2 expression: 2 2a − [3{2(4a − b) − 5(a + 2b)} + 4a]. = x 2+ 1 − 5 y 1 + 2 − 3 2 2 2 3 2 Removing the innermost brackets gives: 2a − [3{8a − 2b − 5a − 10b} + 4a] = x 0 y− 1 3 Collecting together similar terms gives: 2a − [3{3a − 12b} + 4a] = y − 1 or 1 or 1 3 √3 y Removing the ‘curly’ brackets gives: 1 2a − [9a − 36b + 4a] y 3 Collecting together similar terms gives: Now try the following exercise 2a − [13a − 36b] Exercise 1 Revision of basic operations Removing the square brackets gives: and laws of indices 2a − 13a + 36b = −11a + 36b or 36b − 11a 1. Evaluate 2ab + 3bc − abc when a = 2, Problem 8. Factorize (a) x y − 3x z b = −2 and c = 4. [−16] (b) 4a2 + 16ab3 (c) 3a2b − 6ab2 + 15ab. 2. Find the value of 5 pq2r3 when p = 2 , (a) x y − 3x z = x( y − 3z) 5 (b) 4a2 + 16ab3 = 4a(a + 4b3) q = −2 and r = −1. [−8] (c) 3a2b − 6ab2 + 15ab = 3ab(a − 2b + 5) 3. From 4x − 3y + 2z subtract x + 2y − 3z. Problem 9. Simplify 3c + 2c × 4c + c ÷ 5c − 8c. [3x − 5y + 5z] The order of precedence is division, multiplica- 4. Multiply 2a − 5b + c by 3a + b. tion, addition and subtraction (sometimes remembered [6a2 − 13ab + 3ac − 5b2 + bc] by BODMAS). Hence 5. Simplify (x2 y3z)(x3 yz2) and evaluate when x = 1 , y = 2 and z = 3. [x 5 y4 z3, 13 1 ] 2 2 6. Evaluate (a 3 bc−3)(a 1 b− 1 c) when a = 3, 2 2 2 b = 4 and c = 2. [±4 1 ] 2 a2b + a3b 1+a 7. Simplify a2b2 b (a 3b 1 c− 1 )(a b) 1 2 2 3 8. Simplify √ √ ( a3 b c) 11 1 c− 3 √6 a11 √ a 6 b 3 2 or √ 3b c3