Mathematics for Scientific and Technical Students

608 Pages·1998·16.85 MB·English

by  H. Davies, H.G. Davies, G.A. Hicks

Mathematics for scientific and technical students
Second edition
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Mathematics for
scientific and
technical students
Second edition
H. G. Davies and G. A. Hicks
First published 1975 by Addison Wesley Longman Limited
Second Edition 1998
Published 2014 by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OXI4 4RN
711 Third Avenue, New York, NY 10017, USA
Routledge is an imprint of the Taylor & Francis Group. an informa business
Copyright © 1975, 1998, Taylor & Francis.
The right of H. G. Davies and G. A. Hicks to be identified
as authors of this Work has been asserted by them in
accordance with the Copyright, Design and
Patents Act 1988.
All rights reserved. No part ofthis book may be reprinted or reproduced or
utilised in any form or by any electronic, mechanical, or other means, now
known or hereafter invented, including photocopying and recording, or in any
information storage or retrieval system, without permission in writing from
the publishers.
Notices
Knowledge and best practice in this field are constantly changing. As new
research and experience broaden our understanding, changes in research
methods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and
knowledge in evaluating and using any information, methods, compounds, or
experiments described herein. In using such information or methods they
should be mindful oftheir own safety and the safety of others, including
parties for whom they have a professional responsibility.
To the fullest extent ofthe law, neither the Publisher nor the authors,
contributors, or editors, assume any liability 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.
ISBN 13: 978-0-582-41388-7 (pbk)
British Library Cataloguing-in-Publication Data
A catalogue record for this book is
available from the British Library.
Set by MCS Ltd, Salisbury
Contents
Preface Xl
1 Algebra 1
1.1 Quadratic equations 1
1.2 Equations reducible to quadratic form 4
1.3 Simultaneous equations 5
1.4 Indices 12
1.5 Logarithms 14
1.6 The laws of logarithms 18
1.7 Changing the base of a logarithm 21
1.8 Solution of indicial equations 22
1.9 Transposition of equations containing logarithms and
indices 23
1.10 Partial fractions 25
2 Computation 31
2.1 Introduction 31
2.2 Rounding off numbers 31
2.3 Significant figures 32
2.4 Computational accuracy 32
2.5 Definition of error 33
2.6 Error prediction 35
2.7 Standard form 39
3 Trigonometry 42
3.1 Introduction 42
3.2 Trigonometric ratios 42
3.3 Theorem of Pythagoras 44
3 4 Trigonometric ratios for 0°, 30°, 45°, 60°, 90° 46
3.5 Trigonometric ratios of angles greater than 90° 49
3.6 The sine and cosine rules 55
3.7 Area of a triangle 61
vi Contents
3.8 Trigonometric graphs 63
3.9 Other trigonometric ratios 76
3.10 Trigonometric identities 76
3.11 Trigonometric ratios of compound angles 78
3.12 Double angle formulae 81
3.13 Sum and difference formulae 84
3.14 Trigonometric equations 86
3.15 Inverse trigonometric functions 91
4 Differentiation 95
4.1 Functional notation 95
4.2 Incremental values 96
4.3 Limits 97
4.4 The gradient of a curve 100
4.5 Differentiation 102
4.6 Differentiation of y = pxn, where p, n are constants 105
4.7 Another notation for the differential coefficient 106
4.8 Differentiation of polynomials 107
4.9 Repeated differentiation 108
4.10 Differentiation of a function of a function 109
4.11 Differentiation of the product of two functions 111
4.12 Differentiation of a quotient of two functions 113
4.13 The limit of sin () / () as () ---- 0, () being in radians 114
4.14 The differential coefficient of the trigonometric functions 116
4.15 Differentiation of more difficult trigonometric functions 121
4.16 Differentiation of an implicit function 124
4.17 To differentiate the exponential function 125
4.18 To differentiate the log function 128
4.19 Examples of the differentiation of products and quotients
of mixed functions 130
4.20 Differential equations 131
4.21 The differentiation of the inverse trigonometric functions 133
5 Applications of differentiation 140
5.1 Application to gradients of curves 140
5.2 Velocity of sound in a gas 141
5.3 Velocity and acceleration 141
5.4 Turning points 145
5.5 Practical examples of turning points 150
5.6 Rates of change with time 154
5.7 Partial differentiation 157
5.8 Application of partial differentiation to the estimation of
error 160
Contents vii
6 Integration 166
6.1 The area between a curve and the x-axis 166
6.2 Integration: the method of obtaining the exact area under
a curve 167
6.3 The areas under simple curves 169
6.4 Integration of algebraic functions containing coefficients 174
6.5 Area between a curve and the y-axis 176
6.6 Integration as the reverse of differentiation -
the indefinite integral 177
6.7 Integration of more difficult algebraic functions 179
6.8 Further examples of areas under curves 182
6.9 Volumes of revolution 186
6.10 Volume of a cone 188
6.11 Volume of a sphere 188
6.12 Integration by substitution 190
6.13 Functions which integrate to give the logarithm function 193
6.14 Integration of the trigonometric functions 195
6.15 Integration of the trigonometric functions containing
compound angles of the form (bx + c) 197
6.16 Integration of the squared trigonometric functions 199
6.17 Integration of the exponential function 203
6.18 Integration by partial fractions 204
6.19 Integration of a product of two functions (integration
by parts) 206
J
6.20 To determine In x dx 208
6.21 Integrals involving square roots 209
6.22 The integral J 1/(x2 + a2) dx 210
7 Differential equations 215
7.1 Introduction 215
7.2 Differential equations involving direct integration
(variables separable) 215
7.3 First-order differential equations 220
7.4 Examples of differential equations in science and
engineering 224
7.5 Linear differential equations with constant coefficients 228
7.6 Simple harmonic motion 235
8 Applications of integration 240
8.1 Mean values and root mean square values 240
8.2 Centre of gravity and centre of mass 244
8.3 Centroids 246
8.4 Centroids of more complicated areas and the centroid
of a triangle 249
viii Contents
8.5 The centroid of an area beneath a curve y = fix) 251
8.6 Centroid of an arc and a semicircle 253
8.7 The theorems of Pappus (or Guldinus) 255
8.8 Application of Pappus' theorems 257
8.9 Second moments of area and moments of inertia 262
8.10 Two theorems on second moments 268
8.11 Second moment of area of a circle 273
8.12 Second moment of area of an area beneath a curve
between the limits x = a and x = b 275
8.13 Moment of inertia of a solid of revolution 277
9 Graphs 284
9.1 Representation of data 284
9.2 Cartesian and polar coordinates 284
9.3 The distance between two points 287
904 Straight line graphs 290
9.5 Perpendicular lines (normals) 294
9.6 The locus of a point 296
9.7 The circle 298
9.8 The parabola, ellipse and hyperbola 300
9.9 Equation of the tangent and the normal at a point on
a curve 305
9.10 Quadratic graphs 307
9.11 Cubic graphs 309
9.12 Exponential graphs 311
9.13 Logarithmic graphs 314
9.14 Polar graphs 314
9.15 Laws of experimental data 316
9.16 Logarithmic graph paper 325
9.17 The graphical solution to equations 327
9.18 Curve sketching 331
10 Series 344
10.1 Sequences and series 344
10.2 Arithmetic progression (AP) 344
10.3 Geometric progression (GP) 346
lOA Infinite series 349
10.5 Power series 351
10.6 Binomial series 356
10.7 Applications of the binomial theorem 361
10.8 Exponential series 363
11 Numerical methods 368
11.1 Introduction 368
Contents ix
11.2 Solution of equations with one variable 368
11.3 Evaluation of Nk 372
11.4 Linear simultaneous equations 375
11.5 Difference tables 380
11.6 Numerical integration 383
11.7 Differential equations 387
12 Statistics 393
12.1 Introduction 393
12.2 Sampling 393
12.3 Data presentation 394
12.4 Frequency distribution 394
12.5 Numerical measures of a distribution 399
12.6 Numerical measures of central tendency 400
12.7 Numerical measures of dispersion 404
12.8 Linear regression 409
12.9 Linear correlation 413
13 Probability and sampling 423
13.1 Definition of probability 423
13.2 Scale of probability 424
13.3 Compound probability 424
13.4 Probability distributions 428
13.5 Sampling theory 444
13.6 Statistical quality control 445
14 Complex numbers 452
14.1 Definition of a complex number 452
14.2 Operations on complex numbers 452
14.3 Argand diagram 454
14.4 De Moivre's theorem 460
14.5 Exponential form of the complex number 461
14.6 Applications of complex numbers 463
15 Binary arithmetic, sets, logic and Boolean algebra 470
15.1 Binary number system 470
15.2 Binary counting 470
15.3 Number conversion 471
15.4 Operations on binary numbers 473
15.5 Computer arithmetic 476
15.6 Sets 478
15.7 Algebra of sets 482
15.8 Boolean algebra 487
15.9 Switching circuits 489