Engineering Maths First-Aid Kit

162 Pages·2000·1.861 MB·English

by  Tony Croft

Engineering Maths First-Aid Kit
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Engineering Maths First-Aid Kit
Anthony Croft
Pearson Education Ltd
Edinburgh Gate
Harlow
Essex CM20 2JE
England
and Associated Companies around the World.
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First edition 2000
(cid:1)c Pearson Education Limited 2000
The right of Anthony Croft to be identified as the author of this Work has been asserted by him in
accordance with the Copyright, Designs and Patents Act 1988.
All rights reserved. Permission is hereby given for the material in this publication to be reproduced
for OHP transparencies and student handouts, without express permission of the Publishers, for
educational purposes only.
In all other cases, 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 either the prior written permission of the publisher or a licence permitting
restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90
Tottenham Court Road, London W1P 0LP. This book may not be lent, resold, hired out or otherwise
disposed of by way of trade in any form of binding or cover other than that in which it is published,
without prior consent of the Publishers.
ISBN 0130-87430-2
British Library Cataloguing-in-Publication Data
A catalogue record for this book can be obtained from the British Library
Library of Congress Cataloging-in-Publication Data
Croft, Tony, 1957-
Engineering maths first-aid kit / Anthony Croft.– 1st ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-13-087430-2
1. Mathematics. I. Title.
QA37.2.C72 2000
510–dc21 99-088596
10 9 8 7 6 5 4 3 2 1
04 03 02 01 00
Typeset in Computer Modern by 56.
Printed in Great Britain by Henry Ling Ltd., at the Dorset Press, Dorchester, Dorset.
Contents
1. Arithmetic 1.1.1
1.1 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1
1.2 Powers and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1
1.3 Scientific notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1
1.4 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1
1.5 The modulus of a number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1
2. Algebra 2.1.1
2.1 The laws of indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1
2.2 Negative and fractional powers . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1
2.3 Removing brackets 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1
2.4 Removing brackets 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1
2.5 Factorising simple expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1
2.6 Factorising quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1
2.7 Simplifying fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1
2.8 Addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1
2.9 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1
2.10 Rearranging formulas 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1
2.11 Rearranging formulas 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.1
2.12 Solving linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.1
2.13 Simultaneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.1
2.14 Quadratic equations 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14.1
2.15 Quadratic equations 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.1
2.16 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16.1
2.17 The modulus symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17.1
2.18 Graphical solution of inequalities . . . . . . . . . . . . . . . . . . . . . . . 2.18.1
2.19 What is a logarithm? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.1
2.20 The laws of logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20.1
Contents.1 copyright (cid:1)c Pearson Education Limited, 2000
2.21 Bases other than 10 and e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.21.1
2.22 Sigma notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22.1
2.23 Partial fractions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.23.1
2.24 Partial fractions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24.1
2.25 Partial fractions 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.25.1
2.26 Completing the square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.26.1
2.27 What is a surd? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.27.1
3. Functions, coordinate systems and graphs 3.1.1
3.1 What is a function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1
3.2 The graph of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1
3.3 The straight line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1
3.4 The exponential constant e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1
3.5 The hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1
3.6 The hyperbolic identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1
3.7 The logarithm function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1
3.8 Solving equations involving logarithms and exponentials . . . . . . . . . . . 3.8.1
3.9 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1
4. Trigonometry 4.1.1
4.1 Degrees and radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1
4.2 The trigonometrical ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1
4.3 Graphs of the trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 4.3.1
4.4 Trigonometrical identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1
4.5 Pythagoras’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1
4.6 The sine rule and cosine rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1
5. Matrices and determinants 5.1.1
5.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1
5.2 Cramer’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1
5.3 Multiplying matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1
5.4 The inverse of a 2×2 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1
5.5 The inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1
5.6 Using the inverse matrix to solve equations . . . . . . . . . . . . . . . . . . . 5.6.1
6. Vectors 6.1.1
6.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1
Contents.2 copyright (cid:1)c Pearson Education Limited, 2000
6.2 The scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1
6.3 The vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1
7. Complex numbers 7.1.1
7.1 What is a complex number? . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1
7.2 Complex arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1
7.3 The Argand diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1
7.4 The polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1
7.5 The form r(cosθ+jsinθ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1
7.6 Multiplication and division in polar form . . . . . . . . . . . . . . . . . . . . 7.6.1
7.7 The exponential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1
8. Calculus 8.1.1
8.1 Introduction to differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1
8.2 Table of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1
8.3 Linearity rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1
8.4 Product and quotient rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1
8.5 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1
8.6 Integration as the reverse of differentiation . . . . . . . . . . . . . . . . . . . 8.6.1
8.7 Table of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1
8.8 Linearity rules of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1
8.9 Evaluating definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1
8.10 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10.1
8.11 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.1
8.12 Integration as summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12.1
Contents.3 copyright (cid:1)c Pearson Education Limited, 2000

A few words from the author . . .
Over the past three years I have tried to offer mathematical support to many hundreds of
students in the early stages of their degree programmes in engineering.
On many, many occasions I have found that gaps in mathematical knowledge impede progress
both in engineering mathematics and also in some of the engineering topics that the students are
studying. Sometimes these gaps arise because they have long-since forgotten basic techniques.
Sometimes, for a variety of reasons, they seem to have never met certain fundamentals in their
previousstudies. Whatevertheunderlyingreasons, theonlypracticalremedyistohaveavailable
resources which can quickly get to the heart of the problem, which can outline a technique or
formula or important results, and, importantly, which students can take away with them. This
Engineering Maths First-Aid Kit is my attempt at addressing this need.
I am well aware that an approach such as this is not ideal. What many students need is a
prolonged and structured course in basic mathematical techniques, when all the foundations
can be properly laid and there is time to practice and develop confidence. Piecemeal attempts
at helping a student do not really get to the root of the underlying problem. Nevertheless I
see this Kit as a realistic and practical damage-limitation exercise, which can provide sufficient
sticking plaster to enable the student to continue with the other aspects of their studies which
are more important to them.
I have used help leaflets similar to these in the Mathematics Learning Support Centre at Lough-
borough. They are particularly useful at busy times when I may have just a few minutes to
try to help a student, and I would like to revise a topic briefly, and then provide a few simple
practice exercises. You should realise that these leaflets are not an attempt to put together a
coherent course in engineering mathematics, they are not an attempt to replace a textbook, nor
are they intended to be comprehensive in their treatment of individual topics. They are what I
say – elements of a First-Aid kit.
I hope that some of your students find that they ease the pain!
Tony Croft
December 1999
References.4 copyright (cid:1)c Pearson Education Limited, 2000
☛ ✟
1.1
✡ ✠
Fractions
Introduction
The ability to work confidently with fractions, both number fractions and algebraic fractions, is
an essential skill which underpins all other algebraic processes. In this leaflet we remind you of
how number fractions are simplified, added, subtracted, multiplied and divided.
1. Expressing a fraction in its simplest form
p
In any fraction , say, the number p at the top is called the numerator. The number q at
q
the bottom is called the denominator. The number q must never be zero. A fraction can
always be expressed in different, yet equivalent forms. For example, the two fractions 2 and
6
1 are equivalent. They represent the same value. A fraction is expressed in its simplest form
3
by cancelling any factors which are common to both the numerator and the denominator. You
need to remember that factors are numbers which are multiplied together. We note that
2 1×2
=
6 2×3
and so there is a factor of 2 which is common to both the numerator and the denominator. This
common factor can be cancelled to leave the equivalent fraction 1. Cancelling is equivalent to
3
dividing the top and the bottom by the common factor.
Example
12 is equivalent to 3 since
20 5
12 4×3 3
= =
20 4×5 5
Exercises
1. Express each of the following fractions in its simplest form:
a) 12, b) 14, c) 3, d) 100, e) 7, f) 15, g) 3 .
16 21 6 45 9 55 24
Answers
1. a) 3, b) 2, c) 1, d) 20, e) 7, f) 3 , g) 1.
4 3 2 9 9 11 8
2. Addition and subtraction of fractions
To add two fractions we first rewrite each fraction so that they both have the same denominator.
This denominator is chosen to be the lowest common denominator. This is the smallest
1.1.1 copyright (cid:1)c Pearson Education Limited, 2000