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Year:2021Edition:9

Language:EnglishPages (bibliotech):759759

ISBN:9780367643782, 0367643782, 9780367643799, 0367643790ID:2984463

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‘Bird’s Engineering Mathematics 9th Edition’ covers

a wide range of syllabus requirements. The text is suit-

able for any course involving engineering mathematics,

and in particular for the latest National Certificate and

Diploma courses and City & Guilds syllabuses in

Engineering.

This text will provide a foundation in mathematical

principles, which will enable students to solve math-

ematical, scientific and associated engineering prob-

lem. In addition, the material will provide engineering

applications and mathematical principles necessary for

advancement onto a range of Incorporated Engineer

degree profiles. It is widely recognised that a student’s

ability to use mathematics is a key element in deter-

mining subsequent success. First year undergraduates

who need some remedial mathematics will also find this

book meets their needs.

In Bird’s Engineering Mathematics 9thEdition, chapters

have been re-ordered, examples and problems where

engineering applications occur have been ‘flagged up’,

new multiple-choice questions have been added to each

chapter, the text has been added to and simplified,

together with other minor modifications.

Throughout the text, theory is introduced in each chap-

ter by an outline of essential definitions, formulae, laws

and procedures. The theory is kept to a minimum, for

problem solving is extensively used to establish and

exemplify the theory. It is intended that readers will gain

real understanding through seeing problems solved and

then through solving similar problems themselves.

For clarity, the text is divided into eleven topic areas,

these being: number and algebra, trigonometry, areas

and volumes, graphs, complex numbers, vectors, differ-

ential calculus, integral calculus, differential equations,

further number and algebra and statistics.

Table of contents :

Cover

Half Title

Endorsements

Dedication

Title Page

Copyright Page

Contents

Preface

Section 1: Number and algebra

1. Revision of fractions, decimals and percentages

1.1. Fractions

1.2. Ratio and proportion

1.3. Decimals

1.4. Percentages

2. Indices, engineering notation and metric conversions

2.1. Indices

2.2. Standard form

2.3. Engineering notation and common prefixes

2.4. Metric conversions

2.5. Metric – US/imperial conversions

3. Binary, octal and hexadecimal numbers

3.1. Introduction

3.2. Binary numbers

3.3. Octal numbers

3.4. Hexadecimal numbers

4. Calculations and evaluation of formulae

4.1. Errors and approximations

4.2. Use of calculator

4.3. Conversion tables and charts

4.4. Evaluation of formulae

Revision Test 1

5. Algebra

5.1. Basic operations

5.2. Laws of indices

5.3. Brackets and factorisation

5.4. Fundamental laws and precedence

5.5. Direct and inverse proportionality

6. Further algebra

6.1. Polynomial division

6.2. The factor theorem

6.3. The remainder theorem

7. Partial fractions

7.1. Introduction to partial fractions

7.2. Partial fractions with linear factors

7.3. Partial fractions with repeated linear factors

7.4. Partial fractions with quadratic factors

8. Solving simple equations

8.1. Expressions, equations and identities

8.2. Worked problems on simple equations

8.3. Further worked problems on simple

equations

8.4. Practical problems involving simple

equations

8.5. Further practical problems involving

simple equations

Revision Test 2

9. Transposition of formulae

9.1. Introduction to transposition of

formulae

9.2. Worked problems on transposition of

formulae

9.3. Further worked problems on

transposition of formulae

9.4. Harder worked problems on

transposition of formulae

10. Solving simultaneous equations

10.1. Introduction to simultaneous

equations

10.2. Worked problems on simultaneous

equations in two unknowns

10.3. Further worked problems on

simultaneous equations

10.4. More difficult worked problems on

simultaneous equations

10.5. Practical problems involving

simultaneous equations

11. Solving quadratic equations

11.1. Introduction to quadratic equations

11.2. Solution of quadratic equations by

factorisation

11.3. Solution of quadratic equations by

‘completing the square’

11.4. Solution of quadratic equations by formula

11.5. Practical problems involving quadratic

equations

11.6. The solution of linear and quadratic

equations simultaneously

12. Inequalities

12.1. Introduction in inequalities

12.2. Simple inequalities

12.3. Inequalities involving a modulus

12.4. Inequalities involving quotients

12.5. Inequalities involving square functions

12.6. Quadratic inequalities

13. Logarithms

13.1. Introduction to logarithms

13.2. Laws of logarithms

13.3. Indicial equations

13.4. Graphs of logarithmic functions

Revision Test 3

14. Exponential functions

14.1. Introduction to exponential functions

14.2. The power series for ex

14.3. Graphs of exponential functions

14.4. Napierian logarithms

14.5. Laws of growth and decay

15. Number sequences

15.1. Arithmetic progressions

15.2. Worked problems on arithmetic

progressions

15.3. Further worked problems on arithmetic

progressions

15.4. Geometric progressions

15.5. Worked problems on geometric

progressions

15.6. Further worked problems on geometric progressions

15.7. Combinations and permutations

16. The binomial series

16.1. Pascal’s triangle

16.2. The binomial series

16.3. Worked problems on the binomial series

16.4. Further worked problems on the

binomial series

16.5. Practical problems involving the

binomial theorem

Revision Test 4

Section 2: Trigonometry

17. Introduction to trigonometry

17.1. Trigonometry

17.2. The theorem of Pythagoras

17.3. Trigonometric ratios of acute angles

17.4. Fractional and surd forms of

trigonometric ratios

17.5. Evaluating trigonometric ratios of any

angles

17.6. Solution of right-angled triangles

17.7. Angle of elevation and depression

17.8. Trigonometric approximations for small angles

18. Trigonometric waveforms

18.1. Graphs of trigonometric functions

18.2. Angles of any magnitude

18.3. The production of a sine and cosine wave

18.4. Sine and cosine curves

18.5. Sinusoidal form A sin(!t )

18.6. Waveform harmonics

19. Cartesian and polar co-ordinates

19.1. Introduction

19.2. Changing from Cartesian into polar

co-ordinates

19.3. Changing from polar into Cartesian

co-ordinates

19.4. Use of Pol/Rec functions on calculators

Revision Test 5

20. Triangles and some practical applications

20.1. Sine and cosine rules

20.2. Area of any triangle

20.3. Worked problems on the solution of

triangles and their areas

20.4. Further worked problems on the solution of triangles and their areas

20.5. Practical situations involving

trigonometry

20.6. Further practical situations involving

trigonometry

21. Trigonometric identities and equations

21.1. Trigonometric identities

21.2. Worked problems on trigonometric

identities

21.3. Trigonometric equations

21.4. Worked problems (i) on trigonometric

equations

21.5. Worked problems (ii) on trigonometric

equations

21.6. Worked problems (iii) on trigonometric equations

21.7. Worked problems (iv) on trigonometric equations

22. Compound angles

22.1. Compound angle formulae

22.2. Conversion of a sin!t+b cos !t into

R sin(!t+)

22.3. Double angles

22.4. Changing products of sines and cosines

into sums or differences

22.5. Changing sums or differences of sines

and cosines into products

Revision Test 6

Section 3: Areas and volumes

23. Areas of common shapes

23.1. Introduction

23.2. Properties of quadrilaterals

23.3. Areas of common shapes

23.4. Worked problems on areas of common

shapes

23.5. Further worked problems on areas of

plane figures

23.6. Worked problems on areas of composite

figures

23.7. Areas of similar shapes

24. The circle and its properties

24.1. Introduction

24.2. Properties of circles

24.3. Radians and degrees

24.4. Arc length and area of circles and

sectors

24.5. Worked problems on arc length and area

of circles and sectors

24.6. The equation of a circle

25. Volumes and surface areas of common solids

25.1. Introduction

25.2. Volumes and surface areas of regular

solids

25.3. Worked problems on volumes and

surface areas of regular solids

25.4. Further worked problems on volumes

and surface areas of regular solids

25.5. Volumes and surface areas of frusta of

pyramids and cones

25.6. The frustum and zone of a sphere

25.7. Prismoidal rule

25.8. Volumes of similar shapes

26. Irregular areas and volumes and mean values

of waveforms

26.1. Area of irregular figures

26.2. Volumes of irregular solids

26.3. The mean or average value

of a waveform

Revision Test 7

Section 4: Graphs

27. Straight line graphs

27.1. Introduction to graphs

27.2. The straight line graph

27.3. Practical problems involving straight

line graphs

28. Reduction of non-linear laws to linear form

28.1. Determination of law

28.2. Determination of law involving

logarithms

29. Graphs with logarithmic scales

29.1. Logarithmic scales

29.2. Graphs of the form y = axn

29.3. Graphs of the form y = abx

29.4. Graphs of the form y = aekx

30. Graphical solution of equations

30.1. Graphical solution of simultaneous

equations

30.2. Graphical solution of quadratic equations

30.3. Graphical solution of linear and

quadratic equations simultaneously

30.4. Graphical solution of cubic equations

31. Functions and their curves

31.1. Standard curves

31.2. Simple transformations

31.3. Periodic functions

31.4. Continuous and discontinuous functions

31.5. Even and odd functions

31.6. Inverse functions

Revision Test 8

Section 5: Complex numbers

32. Complex numbers

32.1. Cartesian complex numbers

32.2. The Argand diagram

32.3. Addition and subtraction of complex

numbers

32.4. Multiplication and division of complex

numbers

32.5. Complex equations

32.6. The polar form of a complex number

32.7. Multiplication and division in polar form

32.8. Applications of complex numbers

33. De Moivre’s theorem

33.1. Introduction

33.2. Powers of complex numbers

33.3. Roots of complex numbers

Section 6: Vectors

34. Vectors

34.1. Introduction

34.2. Scalars and vectors

34.3. Drawing a vector

34.4. Addition of vectors by drawing

34.5. Resolving vectors into horizontal and

vertical components

34.6. Addition of vectors by calculation

34.7. Vector subtraction

34.8. Relative velocity

34.9. i, j and k notation

35. Methods of adding alternating waveforms

35.1. Combination of two periodic functions

35.2. Plotting periodic functions

35.3. Determining resultant phasors by drawing

35.4. Determining resultant phasors by the

sine and cosine rules

35.5. Determining resultant phasors by

horizontal and vertical components

35.6. Determining resultant phasors by

complex numbers

Revision Test 9

Section 7: Differential calculus

36. Introduction to differentiation

36.1. Introduction to calculus

36.2. Functional notation

36.3. The gradient of a curve

36.4. Differentiation from first principles

36.5. Differentiation of y = axn by the

general rule

36.6. Differentiation of sine and cosine functions

36.7. Differentiation of eax and ln ax

37. Methods of differentiation

37.1. Differentiation of common functions

37.2. Differentiation of a product

37.3. Differentiation of a quotient

37.4. Function of a function

37.5. Successive differentiation

38. Some applications of differentiation

38.1. Rates of change

38.2. Velocity and acceleration

38.3. Turning points

38.4. Practical problems involving maximum

and minimum values

38.5. Points of inflexion

38.6. Tangents and normals

38.7. Small changes

39. Solving equations by Newton’s method

39.1. Introduction to iterative methods

39.2. The Newton–Raphson method

39.3. Worked problems on the

Newton–Raphson method

40. Maclaurin’s series

40.1. Introduction

40.2. Derivation of Maclaurin’s theorem

40.3. Conditions of Maclaurin’s series

40.4. Worked problems on Maclaurin’s series

Revision Test 10

41. Differentiation of parametric equations

41.1. Introduction to parametric equations

41.2. Some common parametric equations

41.3. Differentiation in parameters

41.4. Further worked problems on

differentiation of parametric equations

42. Differentiation of implicit functions

42.1. Implicit functions

42.2. Differentiating implicit functions

42.3. Differentiating implicit functions

containing products and quotients

42.4. Further implicit differentiation

43. Logarithmic differentiation

43.1. Introduction to logarithmic differentiation

43.2. Laws of logarithms

43.3. Differentiation of logarithmic functions

43.4. Differentiation of further logarithmic

functions

43.5. Differentiation of [f (x)]x

Revision Test 11

Section 8: Integral calculus

44. Standard integration

44.1. The process of integration

44.2. The general solution of integrals of the

form axn

44.3. Standard integrals

44.4. Definite integrals

45. Integration using algebraic substitutions

45.1. Introduction

45.2. Algebraic substitutions

45.3. Worked problems on integration using

algebraic substitutions

45.4. Further worked problems on integration

using algebraic substitutions

45.5. Change of limits

46. Integration using trigonometric substitutions

46.1. Introduction

46.2. Worked problems on integration of

sin2 x, cos2 x, tan2 x and cot2 x

46.3. Worked problems on integration of

powers of sines and cosines

46.4. Worked problems on integration of

products of sines and cosines

46.5. Worked problems on integration using the sin θ substitution

46.6. Worked problems on integration using the tan θ substitution

Revision Test 12

47. Integration using partial fractions

47.1. Introduction

47.2. Integration using partial fractions with

linear factors

47.3. Integration using partial fractions with

repeated linear factors

47.4. Integration using partial fractions with

quadratic factors

48. The t =tanθ/2 substitution

48.1. Introduction

48.2. Worked problems on the t = tan 2 substitution

48.3. Further worked problems on the t = tan 2 substitution

49. Integration by parts

49.1. Introduction

49.2. Worked problems on integration by parts

49.3. Further worked problems on integration

by parts

50. Numerical integration

50.1. Introduction

50.2. The trapezoidal rule

50.3. The mid-ordinate rule

50.4. Simpson’s rule

50.5. Accuracy of numerical integration

51. Areas under and between curves

51.1. Area under a curve

51.2. Worked problems on the area under a curve

51.3. Further worked problems on the area

under a curve

51.4. The area between curves

Revision Test 13

52. Mean and root mean square values

52.1. Mean or average values

52.2. Root mean square values

53. Volumes of solids of revolution

53.1. Introduction

53.2. Worked problems on volumes of solids

of revolution

53.3. Further worked problems on volumes of

solids of revolution

54. Centroids of simple shapes

54.1. Centroids

54.2. The first moment of area

54.3. Centroid of area between a curve and the

x-axis

54.4. Centroid of area between a curve and the

y-axis

54.5. Worked problems on centroids of simple

shapes

54.6. Further worked problems on centroids of

simple shapes

54.7. Theorem of Pappus

55. Second moments of area

55.1. Second moments of area and radius of

gyration

55.2. Second moment of area of regular

sections

55.3. Parallel axis theorem

55.4. Perpendicular axis theorem

55.5. Summary of derived results

55.6. Worked problems on second moments of

area of regular sections

55.7. Worked problems on second moments of

area of composite areas

Revision Test 14

Section 9: Differential equations

56. Introduction to differential equations

56.1. Family of curves

56.2. Differential equations

56.3. The solution of equations of the form

dydx= f (x)

56.4. The solution of equations of the form

dydx=f(y)

56.5. The solution of equations of the form

dydx=f(x)f(y)

Revision Test 15

Section 10: Further number and algebra

57. Boolean algebra and logic circuits

57.1. Boolean algebra and switching circuits

57.2. Simplifying Boolean expressions

57.3. Laws and rules of Boolean algebra

57.4. De Morgan’s laws

57.5. Karnaugh maps

57.6. Logic circuits

57.7. Universal logic gates

58. The theory of matrices and determinants

58.1. Matrix notation

58.2. Addition, subtraction and multiplication

of matrices

58.3. The unit matrix

58.4. The determinant of a 2 by 2 matrix

58.5. The inverse or reciprocal of a 2 by 2

matrix

58.6. The determinant of a 3 by 3 matrix

58.7. The inverse or reciprocal of a 3 by 3

matrix

59. The solution of simultaneous equations by

matrices and determinants

59.1. Solution of simultaneous equations by

matrices

59.2. Solution of simultaneous equations by

determinants

59.3. Solution of simultaneous equations

using Cramers rule

59.4. Solution of simultaneous equations

using the Gaussian elimination method

Revision Test 16

Section 11: Statistics

60. Presentation of statistical data

60.1. Some statistical terminology

60.2. Presentation of ungrouped data

60.3. Presentation of grouped data

61. Mean, median, mode and standard deviation

61.1. Measures of central tendency

61.2. Mean, median and mode for

discrete data

61.3. Mean, median and mode for

grouped data

61.4. Standard deviation

61.5. Quartiles, deciles and percentiles

62. Probability

62.1. Introduction to probability

62.2. Laws of probability

62.3. Worked problems on probability

62.4. Further worked problems on probability

62.5. Permutations and combinations

62.6. Bayes’ theorem

Revision Test 17

63. The binomial and Poisson distribution

63.1. The binomial distribution

63.2. The Poisson distribution

64. The normal distribution

64.1. Introduction to the normal distribution

64.2. Testing for a normal distribution

Revision Test 18

65. Linear correlation

65.1. Introduction to linear correlation

65.2. The Pearson product-moment formula

for determining the linear correlation

coefficient

65.3. The significance of a coefficient of

correlation

65.4. Worked problems on linear correlation

66. Linear regression

66.1. Introduction to linear regression

66.2. The least-squares regression

66.3. Worked problems on linear regression

67. Sampling and estimation theories

67.1. Introduction

67.2. Sampling distributions

67.3. The sampling distribution of the means

67.4. The estimation of population parameters

based on a large sample size

67.5. Estimating the mean of a population

based on a small sample size

Revision Test 19

List of essential formulae

Answers to Practice Exercises

Index

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