engineering mathematics pocket book fourth edition pdf

Engineering Mathematics Pocket Book Fourth Edition John Bird... 3.6 Solving equations by iterative methods 58 4.3 Volumes and surface areas of regular solids 82 4.4 Volumes and surfac

Engineering Mathematics Pocket Book

Fourth Edition

John Bird

Engineering Mathematics Pocket Book

Fourth edition

FIMA, FIET, MIEE, FIIE, FCollT

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

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

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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.

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ISBN: 978-0-7506-8153-7

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Printed and bound in United Kingdom

08 09 10 11 12 10 9 8 7 6 5 4 3 2 1

3.6 Solving equations by iterative methods 58

4.3 Volumes and surface areas of regular solids 82 4.4 Volumes and surface areas of frusta of pyramids and cones 88

4.5 The frustum and zone of a sphere 92 4.6 Areas and volumes of irregular figures and solids 95 4.7 The mean or average value of a waveform 101

5.1 Types and properties of angles 105

5.3 Introduction to trigonometry 108 5.4 Trigonometric ratios of acute angles 1095.5 Evaluating trigonometric ratios 110 5.6 Fractional and surd forms of trigonometric ratios 112 5.7 Solution of right-angled triangles 113 5.8 Cartesian and polar co-ordinates 116 5.9 Sine and cosine rules and areas of any triangle 119 5.10 Graphs of trigonometric functions 124

5.12 Sine and cosine waveforms 127 5.13 Trigonometric identities and equations 1345.14 The relationship between trigonometric and hyperbolic functions 139

8.3 Polar form 209 8.4 Applications of complex numbers 211

9.1 Addition, subtraction and multiplication of matrices 217 9.2 The determinant and inverse of a 2 by 2 matrix 218 9.3 The determinant of a 3 by 3 matrix 220 9.4 The inverse of a 3 by 3 matrix 221 9.5 Solution of simultaneous equations by matrices 223 9.6 Solution of simultaneous equations by determinants 226 9.7 Solution of simultaneous equations using Cramer’s rule 2309.8 Solution of simultaneous equations using Gaussian

elimination 232

10.1 Boolean algebra and switching circuits 23410.2 Simplifying Boolean expressions 238 10.3 Laws and rules of Boolean algebra 239

11 Differential Calculus and its Applications 258

11.1 Common standard derivatives 258

11.4 Successive differentiation 262 11.5 Differentiation of hyperbolic functions 263 11.6 Rates of change using differentiation 264 11.7 Velocity and acceleration 265

11.10 Small changes using differentiation 272

11.12 Differentiating implicit functions 276 11.13 Differentiation of logarithmic functions 279 11.14 Differentiation of inverse trigonometric functions 281 11.15 Differentiation of inverse hyperbolic functions 284

11.18 Rates of change using partial differentiation 293 11.19 Small changes using partial differentiation 29411.20 Maxima, minima and saddle points of functions of

two variables 295

12 Integral Calculus and its Applications 303

12.15 Theorem of Pappus 354 12.16 Second moments of area 359

13.1 The solution of equations of the form dy

dxf(x) 366 13.2 The solution of equations of the form dy

dxf(y) 367 13.3 The solution of equations of the form dy

dxf(x).f(y) 368 13.4 Homogeneous first order differential equations 371 13.5 Linear first order differential equations 373 13.6 Second order differential equations of the form

equations 394 13.10 Solution of partial differential equations 405

14.1 Presentation of ungrouped data 416 14.2 Presentation of grouped data 420

14.3 Measures of central tendency 424 14.4 Quartiles, deciles and percentiles 429

15.1 Standard Laplace transforms 472 15.2 Initial and final value theorems 477 15.3 Inverse Laplace transforms 480 15.4 Solving differential equations using Laplace transforms 48315.5 Solving simultaneous differential equations using

16.1 Fourier series for periodic functions of period 2 π 492 16.2 Fourier series for a non-periodic function over range 2 π 496

16.4 Half range Fourier series 501 16.5 Expansion of a periodic function of period L 504 16.6 Half-range Fourier series for functions defined over range L 508 16.7 The complex or exponential form of a Fourier series 511 16.8 A numerical method of harmonic analysis 518 16.9 Complex waveform considerations 522

Index 525

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 formulae, definitions, tables and general information needed during their studies and/or work situation – a handy book to have on the bookshelf to delve into as the need arises

In this 4th edition, the text has been re-designed to make tion easier to access Essential theory, formulae, definitions, laws and procedures are stated clearly at the beginning of each section, and then it is demonstrated how to use such information in practice The text is divided, for convenience of reference, into sixteen main chapters embracing engineering conversions, constants and sym-bols, some algebra topics, some number topics, areas and volumes, geometry and trigonometry, graphs, vectors, complex numbers, matrices and determinants, Boolean algebra and logic circuits, differ-ential and integral calculus and their applications, differential equa-tions, statistics and probability, Laplace transforms and Fourier series

informa-To aid understanding, over 500 application examples have been included, together with over 300 line diagrams

The text assumes little previous knowledge and is suitable for a wide range of courses of study It will be particularly useful for stu-dents studying mathematics within National and Higher National Technician Certificates and Diplomas, GCSE and A levels, for Engineering Degree courses, and as a reference for those in the engineering industry

John Bird Royal Naval School of Marine Engineering, HMS Sultan, formerly University of Portsmouth

and Highbury College, Portsmouth

1 Engineering Conversions, Constants and Symbols

Electric current ampere, A

Thermodynamic temperature kelvin, K

Luminous intensity candela, cd

Amount of substance mole, mol

SI supplementary units

Plane angle radian, rad

Solid angle steradian, sr

Derived units

Electric capacitance farad, F

Electric charge coulomb, C

Electric conductance siemens, S

Electric potential difference volts, V

Electrical resistance ohm, Ω

Magnetic flux density tesla, T

Some other derived units not having special names

Acceleration metre per second squared, m/s 2

Angular velocity radian per second, rad/s

Current density ampere per metre squared, A/m 2

Density kilogram per cubic metre, kg/m 3

Dynamic viscosity pascal second, Pa s

Electric charge density coulomb per cubic metre, C/m 3

Electric field strength volt per metre, V/m

Energy density joule per cubic metre, J/m 3

Heat capacity joule per Kelvin, J/K

Heat flux density watt per square metre, W/m 3

Kinematic viscosity square metre per second, m 2 /s

Luminance candela per square metre, cd/m 2

Magnetic field strength ampere per metre, A/m

Moment of force newton metre, Nm

Permeability henry per metre, H/m

Permittivity farad per metre, F/m

Specific volume cubic metre per kilogram, m3/kg Surface tension newton per metre, N/m

Thermal conductivity watt per metre Kelvin, W/(mK) Velocity metre per second, m/s 2

1.4 Some physical and mathematical constants

Below are listed some physical and mathematical constants, each stated correct to 4 decimal places, where appropriate

Quantity Symbol Value

Speed of light in a

vacuum

c 2.9979  10 8 m/s Permeability of free

c

2 0

4  7.2974  10  3

Coulomb force

constant ke 8.9875  10 9 Nm 2/C2 Gravitational constant G 6.6726  10  11m3/kg s 2 Atomic mass unit u 1.6605  10  27 kg

Rest mass of electron m e 9.1094  10  31 kg

Rest mass of proton m p 1.6726  10  27 kg

Rest mass of neutron m n 1.6749  10  27 kg

Micron μm 10  6 m

Characteristic

impedance of vacuum

Z o 376.7303Ω

Astronomical constants

Mass of earth m E 5.976  10 24 kg Radius of earth R E 6.378  10 6 m Gravity of earth’s surface g 9.8067 m/s 2

Mass of sun M 1.989  10 30 kg Radius of sun R 6.9599  10 8 m Solar effective temperature Te 5800 K

Luminosity of sun L 3.826  10 26 W Astronomical uni t AU 1.496  10 11 m Parsec pc 3.086  10 16 m Jansky Jy 10  26 W/m 2 HZ Tropical year 3.1557  10 7 s Standard atmosphere atm 101325 Pa

infinity 

smaller than or equal to

larger than or equal to

much smaller than

n’th root of a

n

n or d f/dx or f (x)n 2 n

partial differential coefficient of

f(x, y, …) w.r.t x when y, … are held

logarithm to the base a of x loga X

common logarithm of x lg x or log10x

inverse tangent of x tan  1 x or arctan x

inverse secant of x sec  1 x or arcsec x

inverse cosecant of x cosec  1 x or arccosec x

inverse cotangent of x cot  1 x or arccot x

hyperbolic sine of x sinh x

hyperbolic cosine of x cosh x

hyperbolic tangent of x tanh x

hyperbolic secant of x sech x

hyperbolic cosecant of x cosech x

hyperbolic cotangent of x coth x

inverse hyperbolic sine of x sinh  1 x or arsinh x

inverse hyperbolic cosine of x cosh  1 x or arcosh x

inverse hyperbolic tangent of x tanh  1 x or artanh x

inverse hyperbolic secant of x sech  1 x or arsech x

inverse hyperbolic cosecant of x cosech  1 x or arcosech x

inverse hyperbolic cotangent of x coth  1 x or arcoth x

scalar product of vectors A and B A • B

vector product of vectors A and B A  B

1.6 Symbols for physical quantities

(a) Space and time

angle (plane angle) α , β , γ , θ , φ , etc

acceleration, du

acceleration of free fall g

speed of light in a vacuum c

second moment of area I a

second polar moment of area I p

heat; quantity of heat Q, q

work; quantity of work W, w

(e) Electricity and magnetism

Electric charge; quantity of electricity Q

surface charge density σ

electric field strength E

electric current density J, j

magnetic field strength H

absorption factor, absorptance α

reflexion factor, reflectance ρ

transmission factor, transmittance τ

linear extinction coefficient μ

linear absorption coefficient a

speed of longitudinal waves c l

speed of transverse waves c

molar Helmholtz function A m

molar Gibbs function G m

mole fraction of substance B x B

mass fraction of substance B w B

volume fraction of substance B φ B

molality of solute B m B

amount of substance concentration of solute B c B

chemical potential of substance B μ B

absolute activity of substance B λ B

partial pressure of substance B in a gas mixture p B

fugacity of substance B in a gas mixture f B

relative activity of substance B α B

activity coefficient (mole fraction basis) f B

activity coefficient (molality basis) γ B

activity coefficient (concentration basis) y B

osmotic coefficient φ , g

surface concentration Γ

average velocity c, u c u, 0, 0

average speed c , u c u, , most probable speed c uˆ , ˆ

molecular attraction energy ε

interaction energy between molecules i and j φ ij , V ij

distribution function of speeds f(c)

dipole moment of molecule p, μ

quadrupole moment of molecule Θ

first radiation constant c 1

second radiation constant c 2

rotational quantum number J, K

vibrational quantum number v

(j) Atomic and nuclear physics

nucleon number; mass number A

atomic number; proton number Z

(rest) mass of atom m a

unified atomic mass constant m

(rest) mass of electron m e (rest) mass of proton m p (rest) mass of neutron m n elementary charge (of protons) e

orbital angular momentum quantum number L, l 1 spin angular momentum quantum number S, s 1 total angular momentum quantum number J, j 1 nuclear spin quantum number I, J hyperfine structure quantum number F principal quantum number n, n 1 magnetic quantum number M, m 1 fine structure constant α

decay constant λ

2 1 2

, t

spin-lattice relaxation time T 1

spin-spin relaxation time T 2

indirect spin-spin coupling J

(k) Nuclear reactions and ionising radiations

internal conversion coefficient α

linear attenuation coefficient μ , μ 1

atomic attenuation coefficient μ

mass attenuation coefficient μ m

linear stopping power S, S 1

atomic stopping power S a

recombination coefficient α

2.1 Polynomial division

Application: Divide 2x 2  x  3 by x  1

2x 2  x  3 is called the dividend and x  1 the divisor The usual

layout is shown below with the dividend and divisor both arranged

in descending powers of the symbols

xx

−

Dividing the first term of the dividend by the first term of the sor, i.e 2x x2/ gives 2x, which is placed above the first term of the dividend as shown The divisor is then multiplied by 2x, i.e 2x(x  1)  2x 2  2x, which is placed under the dividend as shown Subtracting gives 3x  3

divi-The process is then repeated, i.e the first term of the divisor, x, is divided into 3x, giving  3, which is placed above the dividend as shown Then 3(x  1)  3x  3 which is placed under the 3x  3.The remainder, on subtraction, is zero, which completes the process

Thus, (2x 2  x  3) ÷ (x  1)  (2x  3)

Application: Divide (x 2  3x  2) by (x  2)

x

xx

2.2 The factor theorem

A factor of (x  a) in an equation corresponds to a root of

If f(3)  0, then (x  3) is a factor – from the factor theorem

We have a choice now We can divide x 3  7x  6 by (x  3) or we could continue our ‘trial and error ’ by substituting further values for

x in the given expression – and hope to arrive at f(x)  0

Let us do both ways Firstly, dividing out gives:

Therefore let us try some negative values for x

f(  1)  (  1) 3  7( 1)  6  0; hence (x  1) is a factor (as shown above)

Also f(  2)  (  2) 3  7(  2)  6  0; hence (x  2) is a factor

To solve x 3  7x  6  0, we substitute the factors, i.e

(x3)(x1)(x 2) 0

from which, x  3, x   1 and x  2

Note that the values of x, i.e 3,1 and 2, are all factors of the constant term, i.e the 6 This can give us a clue as to what values of

x we should consider

2.3 The remainder theorem

If (ax 2  bx  c) is divided by (x  p), the remainder will be

hence the remainder is 3(2) 2  (  4)(2)  5  12  8  5  9

We can check this by dividing (3x 2  4x  5) by (x  2) by long division:

which means that (x  1) is a factor of (2x 2  x  3)

In this case, the other factor is (2x  3),

i.e (2x 2  x  3)  (x  1)(2x  3)

Application: When (3x 3  2x 2  x  4) is divided by (x  1), find the remainder

The remainder is ap 3  bp 2  cp  d (where a  3, b  2, c   1,

d  4 and p  1), i.e the remainder is:

Comparisons show that A, B, C and D are 2, 8, 1 and 2 respectively

A fraction written in the general form is called a continued tion and the integers A, B, C and D are called the quotients of the

frac-continued fraction The quotients may be used to obtain closer and

closer approximations, called convergents

A tabular method may be used to determine the convergents of a fraction:

55

The quotients 2, 8, 1 and 2 are written in cells a2, a3, a4 and a5 with cell a1 being left empty

The fraction 01 is always written in cell b1

The reciprocal of the quotient in cell a2 is always written in cell b2, i.e.1

55 These approximations to fractions are used to obtain practical ratios

for gearwheels or for a dividing head (used to give a required

Application: The height s metres of a mass projected vertically

upwards at time t seconds is sut1gt

2

2 Determine how long the mass will take after being projected to reach a height of 16 m (a) on the ascent and (b) on the descent, when u  30 m/s and

g  9.81 m/s 2

Using the quadratic formula:

Application: A shed is 4.0 m long and 2.0 m wide A concrete

path of constant width is laid all the way around the shed and the area of the path is 9.50 m 2 Calculate its width, to the nearest centimetre

Figure 2.1 shows a plan view of the shed with its surrounding path