Bird's engineering mathematics

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