Engineering Mathematics Through Applications

944 Pages·2011·10.85 MB·English

by  Kuldeep Singh

Engineering
mathematics
through
applications
Second edition
KULDEEP SINGH
Senior Lecturer in Mathematics
School of Physics, Astronomy and Mathematics
University of Hertfordshire
© Kuldeep Singh 2003, 2011
All rights reserved. No reproduction, copy or transmission of this
publication may be made without written permission.
No portion of this publication may be reproduced, copied or transmitted
save with written permission or in accordance with the provisions of the
Copyright, Designs and Patents Act 1988, or under the terms of any licence
permitting limited copying issued by the Copyright Licensing Agency,
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Any person who does any unauthorized act in relation to this publication
may be liable to criminal prosecution and civil claims for damages.
The author has asserted his right to be identified
as the author of this work in accordance with the Copyright,
Designs and Patents Act 1988.
First edition 2003
This edition first published 2011 by
PALGRAVE MACMILLAN
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registered in England, company number 785998, of Houndmills, Basingstoke,
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Palgrave® and Macmillan® are registered trademarks in the United States,
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ISBN 978–0–230–27479–2 paperback
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Printed and bound in Great Britain by
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Acknowledgements
The author and publishers would like to thank the following for the use of copyright material:
Maplesoft Inc.
Cambridge University Press for the table on page 918, taken from J. C. P. Miller & F. C. Powell,
The Cambridge Elementary Mathematical Tables, 2nd edn, 1979. Reproduced with permission.
The Engineering Council.
The author and publishers are grateful to the following for permission to reproduce questions
from past examination papers:
Jay Abramson, Arizona State University, USA; Eric Bahuaud, Stanford University, USA; Fabrice
Baudoin, Purdue University, USA; David Bayer, Barnard College, Columbia University, New
York, USA; Maretta Brennan, Cork Institute of Technology, Ireland; Dietrich Burbulla,
University of Toronto, Canada; Michael Chung, University of Aberdeen, UK; Francis Coghlan,
University of Manchester, UK; Stephen Gourley, University of Surrey, UK; Rhian Green,
University of Loughborough, UK; Alexander Hulpke, Colorado State University, USA; Jennifer
Kloke, Stanford University, USA; Duane Kouba, University of California, Davis, USA; Roger
Luther, University of Sussex, UK; Fiona Message, University of Portsmouth, UK; Wayne Nagata,
University of British Columbia, Canada; Erhard Neher, University of Ottawa, Canada; Pascal
O’Connor, Cork Institute of Technology, Ireland; Sean O’Rourke, University of California,
Davis, USA; John Parkinson, University of Manchester, UK; Nicolai Reshetikhin, University of
California, Berkeley, USA; Alyssa Sankey, University of New Brunswick, Canada; Marshall
Slemrod, University of Wisconsin, USA; Peter Sollich, King’s College London, UK; Toby
Stafford, University of Manchester, UK; Colin Steele, University of Manchester, UK; Shannon
Sullivan, Memorial University of Newfoundland, Canada; Yi Sun, North Carolina State
University, USA; Joachim Vogt, Jacobs University, Germany; Louise Walker, University of
Manchester, UK; Joe Ward, University of Loughborough, UK; Jack Williams, University of
Manchester, UK; Francis Wright, Queen Mary, University of London, UK.
Dedication
— To Bibi Chanan Kaur
iii
Summary of Contents
Acknowledgements iii
Note to the Student ix
Preface to Second Edition xi
Important Formulae and Methods xiii
INTRODUCTION Arithmetic for Engineers 1
1 Engineering Formulae 52
2 Visualizing Engineering Formulae 100
3 Functions in Engineering 136
4 Trigonometry and Waveforms 171
5 Logarithmic, Exponential and Hyperbolic Functions 235
6 Differentiation 271
7 Engineering Applications of Differentiation 326
8 Integration 399
9 Engineering Applications of Integration 472
10 Complex Numbers 513
11 Matrices 560
12 Vectors 636
13 First Order Differential Equations 672
14 Second Order Linear Differential Equations 730
15 Partial Differentiation 772
16 Probability and Statistics 806
Solutions 888
Appendix: Standard Normal Distribution Table 918
Index 919
iv
Contents
Acknowledgements iii
Note to the Student ix
Preface to Second Edition xi
Important Formulae and Methods xiii
INTRODUCTION Arithmetic for Engineers 1
(cid:2) Whole numbers 2
(cid:2) Indices 7
(cid:2) Numbers 12
(cid:2) Fractions 16
(cid:2) Arithmetic of fractions 18
(cid:2) Decimals 24
(cid:2) Powers of 10 28
(cid:2) Conversion 38
(cid:2) Arithmetical operations 40
(cid:2) Percentages 43
(cid:2) Ratios 48
1 Engineering Formulae 52
(cid:2) Substitution and transposition 53
(cid:2) Transposing engineering formulae 59
(cid:2) Indices 66
(cid:2) Dimensional analysis 69
(cid:2) Expansion of brackets 73
(cid:2) Factorization 78
(cid:2) Quadratic equations 86
(cid:2) Simultaneous equations 90
2 Visualizing Engineering Formulae 100
(cid:2) Graphs 101
(cid:2) Applications of graphs 106
(cid:2) Quadratic graphs 110
(cid:2) Quadratics revisited 115
(cid:2) Further graphs 121
(cid:2) Binomial expansion 129
3 Functions in Engineering 136
(cid:2) Concepts of functions 137
(cid:2) Inverse functions 141
v
vi (cid:2)Contents
(cid:2) Graphs of functions 145
(cid:2) Combinations of functions 153
(cid:2) Limits of functions 159
(cid:2) Modulus function 166
4 Trigonometry and Waveforms 171
(cid:2) Trigonometric functions 172
(cid:2) Angles and graphs 181
(cid:2) Trigonometric equations 187
(cid:2) Trigonometric rules 195
(cid:2) Radians 201
(cid:2) Wave theory 204
(cid:2) Trigonometric identities 212
(cid:2) Applications of identities 219
(cid:2) Conversion 223
5 Logarithmic, Exponential and Hyperbolic Functions 235
(cid:2) Indices revisited 236
(cid:2) The exponential function 239
(cid:2) The logarithmic function 245
(cid:2) Applications of logarithms 254
(cid:2) Hyperbolic functions 259
6 Differentiation 271
(cid:2) The derivative 272
(cid:2) Derivatives of functions 282
(cid:2) Chain rule revisited 290
(cid:2) Product and quotient rules 296
(cid:2) Higher derivatives 302
(cid:2) Parametric differentiation 308
(cid:2) Implicit and logarithmic differentiation 315
7 Engineering Applications of Differentiation 326
(cid:2) Curve sketching 327
(cid:2) Optimization problems 339
(cid:2) First derivative test 345
(cid:2) Applications to kinematics 349
(cid:2) Tangents and normals 355
(cid:2) Series expansion 358
(cid:2) Binomial revisited 369
(cid:2) An introduction to infinite series 375
(cid:2) Numerical solution of equations 389
8 Integration 399
(cid:2) Introduction to integration 400
(cid:2) Rules of integration 407
(cid:2)Contents vii
(cid:2) Integration by substitution 414
(cid:2) Applications of integration 421
(cid:2) Integration by parts 432
(cid:2) Algebraic fractions 440
(cid:2) Integration of algebraic fractions 450
(cid:2) Integration by substitution revisited 455
(cid:2) Trigonometric techniques for integration 459
9 Engineering Applications of Integration 472
(cid:2) Trapezium rule 473
(cid:2) Further numerical integration 481
(cid:2) Engineering applications 489
(cid:2) Applications in mechanics 499
(cid:2) Miscellaneous applications of integration 503
10 Complex Numbers 513
(cid:2) Arithmetic of complex numbers 514
(cid:2) Representation of complex numbers 525
(cid:2) Multiplication and division in polar form 532
(cid:2) Powers and roots of complex numbers 538
(cid:2) Exponential form of complex numbers 548
11 Matrices 560
(cid:2) Manipulation of matrices 561
(cid:2) Applications 574
(cid:2) 3(cid:2)3 matrices 585
(cid:2) Gaussian elimination 598
(cid:2) Linear equations 606
(cid:2) Eigenvalues and eigenvectors 614
(cid:2) Diagonalization 624
12 Vectors 636
(cid:2) Vector representation 637
(cid:2) Vectors in Cartesian co-ordinates 645
(cid:2) Three-dimensional vectors 649
(cid:2) Scalar products 655
(cid:2) Vector products 662
13 First Order Differential Equations 672
(cid:2) Solving differential equations 673
(cid:2) Using the integrating factor 683
(cid:2) Applications to electrical principles 689
(cid:2) Further engineering applications 695
(cid:2) Euler’s numerical method 703
(cid:2) Improved Euler’s method 711
(cid:2) Fourth order Runge–Kutta 718
viii (cid:2)Contents
14 Second Order Linear Differential
Equations 730
(cid:2) Homogeneous differential equations 731
(cid:2) Engineering applications 738
(cid:2) Non-homogeneous (inhomogeneous) differential equations 746
(cid:2) Particular solutions 759
15 Partial Differentiation 772
(cid:2) Partial derivatives 773
(cid:2) Applications 784
(cid:2) Optimization 792
16 Probability and Statistics 806
(cid:2) Data representation 807
(cid:2) Data summaries 816
(cid:2) Probability rules 828
(cid:2) Permutations and combinations 840
(cid:2) Binomial distribution 845
(cid:2) Properties of discrete random variables 853
(cid:2) Properties and applications of continuous random variables 862
(cid:2) Normal distribution 875
Solutions 888
Appendix: Standard Normal Distribution Table 918
Index 919
Note to the Student
This book is ideally suited for anyone doing mathematics as part of their engineering or science
undergraduate studies. Very little mathematical knowledge is assumed.
The heart of the book is the wealth of engineering examples drawn from a wide range of
disciplines such as aerospace, building services, civil, control, electrical, manufacturing,
mechanical, etc. In this respect the book is unique. The inclusion of these engineering
disciplines will show from the outset the many applications of the mathematics you are
studying. Understanding why you need to learn the mathematics and using these
mathematical principles in real engineering examples are a real motivating factor.
These examples do not assume prior engineering knowledge. However you will be asked to reflect
on your final answer, i.e. how does your result relate to the problem? Reflection is a good way of
learning mathematics and therefore frequent use of a question and answer format is made
throughout the book. A question mark icon ? represents the questions. The question mark icon
will help you to stop and think rather than pressing on and reading the answer on the next line.
The key to learning mathematics is to do exercises. If you want to become familiar with
mathematical language and understand the theory there is no substitute for practice. There
are exercises associated with each section and a miscellaneous exercise at the end of each
chapter. The exercises are an integral part of the book and not just a bolt on. Moreover every
section begins with a list of objectives and ends with a summary. The chapter also begins with
a list of objectives. Doing the exercises and checking your answers are the only way you can
meet these objectives.
Complete solutions to all exercises are provided on the book’s website at
www.palgrave.com/engineering/singh. This makes the book suitable for students working by
themselves or on a distance learning course, as well as for lecture-based courses. These complete
solutions provide a thorough step-by-step guide through each problem. Also on the website is
a link through to dozens of test questions which offer immediate feedback and scores, an online
glossary of key terms, some additional sections and an e-index to help you find the right page
in the book immediately.
The proof of mathematical results is kept to a minimum. Instead the emphases of the book
are on you learning by investigating results, observing patterns, visualizing graphs, answering
questions using technology, etc.
You need to understand the first two chapters thoroughly, because basic arithmetic and
algebra are vital ingredients for the remaining chapters. For example, you cannot do any
calculus unless you have a firm footing in basic algebra. The content of these chapters should
eventually become second nature to you. Recent news articles have stated that there are fewer
and fewer students applying for engineering courses at university, which is a serious problem
for industry. One of the reasons is that the mathematical nature of engineering is perceived
as difficult. This book addresses this problem by using an everyday language step-by-step
guide through each example. For example, if an expansion of brackets is carried out at a
particular step in the manipulation, then this is stated as [Expanding] adjacent to the step.
Every formula has been given a reference number. For example, reference number 2.1 means
that it is the first formula in Chapter 2. If a particular formula is used, then it is either stated
in the main text or placed in the footnote on the page so that you do not need to search for it.
ix