دانلود کتاب Bird’s Higher Engineering Mathematics

فهرست مطالب :

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Syllabus guidance
Section A: Number and algebra
1. Algebra
1.1. Introduction
1.2. Revision of basic laws
1.3. Revision of equations
1.4. Polynomial division
1.5. The factor theorem
1.6. The remainder theorem
2. Partial fractions
2.1. Introduction to partial fractions
2.2. Partial fractions with linear factors
2.3. Partial fractions with repeated linear factors
2.4. Partial fractions with quadratic factors
3. Logarithms
3.1. Introduction to logarithms
3.2. Laws of logarithms
3.3. Indicial equations
3.4. Graphs of logarithmic functions
4. Exponential functions
4.1. Introduction to exponential functions
4.2. The power series for e x
4.3. Graphs of exponential functions
4.4. Napierian logarithms
4.5. Laws of growth and decay
4.6. Reduction of exponential laws to linear form
Revision Test 1
5. The binomial series
5.1. Pascal’s triangle
5.2. The binomial series
5.3. Worked problems on the binomial series
5.4. Further worked problems on the binomial series
5.5. Practical problems involving the binomial theorem
6. Solving equations by iterative methods
6.1. Introduction to iterative methods
6.2. The bisection method
6.3. An algebraic method of successive approximations
7. Boolean algebra and logic circuits
7.1. Boolean algebra and switching circuits
7.3. Laws and rules of Boolean algebra
7.2. Simplifying Boolean expressions
7.4. De Morgan’s laws
7.5. Karnaugh maps
7.6. Logic circuits
7.7. Universal logic gates
Revision Test 2
Section B: Geometry and trigonometry
8. Introduction to trigonometry
8.1. Trigonometry
8.2. The theorem of Pythagoras
8.3. Trigonometric ratios of acute angles
8.4. Evaluating trigonometric ratios
8.5. Solution of right-angled triangles
8.6. Angles of elevation and depression
8.7. Sine and cosine rules
8.8. Area of any triangle
8.9. Worked problems on the solution of triangles and finding their areas
8.10. Further worked problems on solving triangles and finding their areas
8.11. Practical situations involving trigonometry
8.12. Further practical situations involving trigonometry
9. Cartesian and polar co-ordinates
9.1. Introduction
9.2. Changing from Cartesian into polar co-ordinates
9.3. Changing from polar into Cartesian co-ordinates
9.4. Use of Pol/Rec functions on calculators
10. The circle and its properties
10.1. Introduction
10.2. Properties of circles
10.3. Radians and degrees
10.4. Arc length and area of circles and sectors
10.5. The equation of a circle
10.6. Linear and angular velocity
10.7. Centripetal force
Revision Test 3
11. Trigonometric waveforms
11.1. Graphs of trigonometric functions
11.2. Angles of any magnitude
11.3. The production of a sine and cosine wave
11.4. Sine and cosine curves
11.5 Sinusoidal form A sin(!t )
11.6. Harmonic synthesis with complex waveforms
12. Hyperbolic functions
12.1. Introduction to hyperbolic functions
12.2. Graphs of hyperbolic functions
12.3. Hyperbolic identities
12.4. Solving equations involving hyperbolic functions
12.5. Series expansions for cosh x and sinh x
13. Trigonometric identities and equations
13.1. Trigonometric identities
13.2. Worked problems on trigonometric identities
13.3. Trigonometric equations
13.4. Worked problems (i) on trigonometric equations
13.5. Worked problems (ii) on trigonometric equations
13.6. Worked problems (iii) on trigonometric equations
13.7. Worked problems (iv) on trigonometric equations
14. The relationship between trigonometric and hyperbolic functions
14.1. The relationship between trigonometric and hyperbolic functions
14.2. Hyperbolic identities
15. Compound angles
15.1. Compound angle formulae
15.2. Conversion of a sin !t + b cos !t into
R sin(!t + )
15.3. Double angles
15.4. Changing products of sines and cosines into sums or differences
15.5. Changing sums or differences of sines and cosines into products
15.6. Power waveforms in a.c. circuits
Revision Test 4
Section C: Graphs
16. Functions and their curves
16.1. Standard curves
16.2. Simple transformations
16.3. Periodic functions
16.4. Continuous and discontinuous functions
16.5. Even and odd functions
16.6. Inverse functions
16.7. Asymptotes
16.8. Brief guide to curve sketching
16.9. Worked problems on curve sketching
17. Irregular areas, volumes and mean values of waveforms
17.1. Areas of irregular figures
17.2. Volumes of irregular solids
17.3. The mean or average value of a waveform
Revision Test 5
Section D: Complex numbers
18. Complex numbers
18.1. Cartesian complex numbers
18.2. The Argand diagram
18.3. Addition and subtraction of complex numbers
18.2. The Argand diagram
18.4. Multiplication and division of complex numbers
18.5. Complex equations
18.6. The polar form of a complex number
18.7. Multiplication and division in polar form
18.8. Applications of complex numbers
19. De Moivre’s theorem
19.1. Introduction
19.2. Powers of complex numbers
19.3. Roots of complex numbers
19.4. The exponential form of a complex number
19.5. Introduction to locus problems
Section E: Matrices and determinants
20. The theory of matrices and determinants
20.1. Matrix notation
20.2. Addition, subtraction and multiplication of matrices
20.3. The unit matrix
20.4. The determinant of a 2 by 2 matrix
20.5. The inverse or reciprocal of a 2 by 2 matrix
20.6. The determinant of a 3 by 3 matrix
20.7. The inverse or reciprocal of a 3 by 3 matrix
21. Applications of matrices and determinants
21.1. Solution of simultaneous equations by matrices
21.2. Solution of simultaneous equations by determinants
21.3. Solution of simultaneous equations using Cramer’s rule
21.4. Solution of simultaneous equations using the Gaussian elimination method
21.5. Stiffness matrix
21.6. Eigenvalues and eigenvectors
Section F: Vector geometry
22. Vectors
22.1. Introduction
22.2. Scalars and vectors
22.3. Drawing a vector
22.4. Addition of vectors by drawing
22.5. Resolving vectors into horizontal and vertical components
22.6. Addition of vectors by calculation
22.7. Vector subtraction
22.8. Relative velocity
22.9. i, j and k notation
23. Methods of adding alternating waveforms
23.1. Combination of two periodic functions
23.2. Plotting periodic functions
23.3. Determining resultant phasors by drawing
23.4. Determining resultant phasors by the sine and cosine rules
23.5. Determining resultant phasors by horizontal and vertical components
23.6. Determining resultant phasors by using complex numbers
24. Scalar and vector products
24.1. The unit triad
24.2. The scalar product of two vectors
24.3. Vector products
24.4. Vector equation of a line
Revision Test 7
Section G: Differential calculus
25. Methods of differentiation
25.1. Introduction to calculus
25.2. The gradient of a curve
25.3. Differentiation from first principles
25.4. Differentiation of common functions
25.5. Differentiation of a product
25.6. Differentiation of a quotient
25.7. Function of a function
25.8. Successive differentiation
26. Some applications of differentiation
26.1. Rates of change
26.2. Velocity and acceleration
26.3. The Newton–Raphson method
26.4. Turning points
26.5. Practical problems involving maximum and minimum values
26.6. Points of inflexion
26.7. Tangents and normals
26.8. Small changes
Revision Test 8
27. Differentiation of parametric equations
27.1. Introduction to parametric equations
27.2. Some common parametric equations
27.3. Differentiation in parameters
27.4. Further worked problems on differentiation of parametric equations
28. Differentiation of implicit functions
28.1. Implicit functions
28.2. Differentiating implicit functions
28.3. Differentiating implicit functions containing products and quotients
28.4. Further implicit differentiation
29. Logarithmic differentiation
29.1. Introduction to logarithmic differentiation
29.2. Laws of logarithms
29.4. Differentiation of further logarithmic functions
29.3. Differentiation of logarithmic functions
29.5. Differentiation of [f(x)]x
Revision Test 9
30. Differentiation of hyperbolic functions
30.1. Standard differential coefficients of hyperbolic functions
30.2. Further worked problems on differentiation of hyperbolic functions
31. Differentiation of inverse trigonometric and hyperbolic functions
31.1. Inverse functions
31.2. Differentiation of inverse trigonometric functions
31.3. Logarithmic forms of inverse hyperbolic functions
31.4. Differentiation of inverse hyperbolic functions
32. Partial differentiation
32.1. Introduction to partial derivatives
32.2. First-order partial derivatives
32.3. Second-order partial derivatives
33. Total differential, rates of change and small changes
33.1. Total differential
33.2. Rates of change
33.3. Small changes
34. Maxima, minima and saddle points for functions of two variables
34.1. Functions of two independent variables
34.2. Maxima, minima and saddle points
34.3. Procedure to determine maxima, minima and saddle points for functions of two variables
34.4. Worked problems on maxima, minima and saddle points for functions of two variables
34.5. Further worked problems on maxima, minima and saddle points for functions of two variables
Revision Test 10
Section H: Integral calculus
35. Standard integration
35.1. The process of integration
35.2. The general solution of integrals of the form axn
35.3. Standard integrals
35.4. Definite integrals
36. Some applications of integration
36.1. Introduction
36.2. Areas under and between curves
36.3. Mean and rms values
36.4. Volumes of solids of revolution
36.5. Centroids
36.6. Theorem of Pappus
36.7. Second moments of area of regular sections
Revision Test 11
37. Maclaurin’s series and limiting values
37.1. Introduction
37.2. Derivation of Maclaurin’s theorem
37.3. Conditions of Maclaurin’s series
37.4. Worked problems on Maclaurin’s series
37.5. Numerical integration using Maclaurin’s series
37.6. Limiting values
38. Integration using algebraic substitutions
38.1. Introduction
38.2. Algebraic substitutions
38.3. Worked problems on integration using algebraic substitutions
38.4. Further worked problems on integration using algebraic substitutions
38.5. Change of limits
39. Integration using trigonometric and hyperbolic substitutions
39.1. Introduction
39.2. Worked problems on integrationof sin2 x, cos2 x, tan2 x and cot2 x
39.3. Worked problems on integration of powers of sines and cosines
39.4. Worked problems on integration of products of sines and cosines
39.5. Worked problems on integration using
the sin  substitution
39.6. Worked problems on integration using the tan substitution
39.7. Worked problems on integration using the sinh substitution
39.8. Worked problems on integration using the cosh substitution
40. Integration using partial fractions
40.1 Introduction
40.2 Integration using partial fractions with linear factors
40.3 Integration using partial fractions with repeated linear factors
40.4 Integration using partial fractions with quadratic factors
41. The t=tan 2 substitution
41.1. Introduction
41.2. Worked problems on the t = tan 2 substitution
41.3. Further problems on the t = tan 2 substitution
Revision Test 12
42. Integration by parts
42.1. Introduction
42.2. Worked problems on integration by parts
42.3. Further worked problems on integration by parts
43. Reduction formulae
43.1. Introduction
43.2. Using reduction formulae for integrals of the form ∫ xn e x dx
43.3. Using reduction formulae forintegrals of the form∫xn cos x dxand∫xn sin x dx
43.4. Using reduction formulae forintegrals of the form∫sinn x dx and∫cosn x dx
43.5. Further reduction formulae
44. Double and triple integrals
44.1. Double integrals
44.2. Triple integrals
45. Numerical integration
45.1. Introduction
45.2. The trapezoidal rule
45.3. The mid-ordinate rule
45.4. Simpson’s rule
45.5. Accuracy of numerical integration
Revision Test 13
Section I: Differential equations
46. Introduction to differential equations
46.1. Family of curves
46.2. Differential equations
46.3. The solution of equations of the form dy dx=f(x)
46.4. The solution of equations of the form dy dx=f( y)
46.5. The solution of equations of theformdydx= f(x)
f( y)
47. Homogeneous first-order differential equations
47.1. Introduction
47.2. Procedure to solve differentialequations of the form P dy dx=Q
47.3. Worked problems on homogeneous first-order differential equations
47.4. Further worked problems on homogeneous first-order differential equations
48. Linear first-order differential equations
48.1. Introduction
48.2. Procedure to solve differential equations of the form dydx+Py=Q
48.3. Worked problems on linear first-order differential equations
48.4. Further worked problems on linear first-order differential equations
49. Numerical methods for first-order differential equations
49.1. Introduction
49.2. Euler’s method
49.3. Worked problems on Euler’s method
49.4. The Euler–Cauchy method
49.5. The Runge–Kutta method
Revision Test 14
50. Second-order differential equations of the form ad2y dx2+bdy dx+cy=0
50.1. Introduction
50.2. Procedure to solve differential equations of the form a d2y dx2 +b dy dx + cy=0
50.3. Worked problems ondifferential equations ofthe form a d2y dx2 +b dy dx + cy = 0
50.4. Further worked problems on practical differential equations of the form a d2y dx2 +b dy dx + cy = 0
51. Second-order differential equations of the form ad2y dx2 + bdy dx+cy=f(x)
51.1. Complementary function and particular integral
51.2. Procedure to solve differential equations of the form a d2y dx2 +b dy dx + cy = f(x)
51.3. Differential equations of the form a d2y dx2 +b dy dx+cy = f(x) where f(x) is a constant orpolynomial
51.4. Differential equations of the form a d2y dx2 + b dy dx+cy = f(x) where f(x) is an exponential function
51.5. Differential equations of the form a d2y dx2 +b dy dx+cy=f(x) where f(x) is a sine or cosine function
51.6. Differential equations of the form a d2y dx2 +b dy dx+cy = f(x) where f(x) is a sum or a product
52. Power series methods of solving ordinary differential equations
52.1. Introduction
52.2. Higher order differential coefficients as series
52.3. Leibniz’s theorem
52.4. Power series solution by the Leibniz–Maclaurin method
52.5. Power series solution by the Frobenius method
52.6. Bessel’s equation and Bessel’s functions
52.7. Legendre’s equation and Legendre polynomials
53. An introduction to partial differential equations
53.1. Introduction
53.2. Partial integration
53.3. Solution of partial differential equations by direct partial integration
53.4. Some important engineering partial differential equations
53.5. Separating the variables
53.6. The wave equation
53.7. The heat conduction equation
53.8. Laplace’s equation
Revision Test 15
Section J: Laplace transforms
54. Introduction to Laplace transforms
54.1. Introduction
54.2. Definition of a Laplace transform
54.3. Linearity property of the Laplace transform
54.4. Laplace transforms of elementary functions
54.5. Worked problems on standard Laplace transforms
55. Properties of Laplace transforms
55.1. The Laplace transform of eat
55.2. Laplace transforms of the form e at f(t)
55.3. The Laplace transforms of derivatives
55.4. The initial and final value theorems
56. Inverse Laplace transforms
56.1. Definition of the inverse Laplace transform
56.2. Inverse Laplace transforms of simple functions
56.3. Inverse Laplace transforms using partial fractions
56.4. Poles and zeros
57. The Laplace transform of the Heaviside function
57.1. Heaviside unit step function
57.2. Laplace transform of H(t – c)
57.3. Laplace transform of H(t – c)
f(t – c)
57.4. Inverse Laplace transforms of Heaviside functions
58. The solution of differential equations using Laplace transforms
58.1. Introduction
58.2. Procedure to solve differential equations by using Laplace transforms
58.3. Worked problems on solving differential equations using Laplace transforms
59. The solution of simultaneous differential equations using Laplace transforms
59.1. Introduction
59.2. Procedure to solve simultaneous differential equations using Laplace transforms
59.3. Worked problems on solving simultaneous differential equations by using Laplace transforms
Revision Test 16
Section K: Fourier series
60. Fourier series for periodic functions of period 2ˇ
60.1. Introduction
60.2. Periodic functions
60.3. Fourier series
60.4. Worked problems on Fourier series of periodic functions of period 2ˇ
61. Fourier series for a non-periodic function over range 2ˇ
61.1. Expansion of non-periodic functions
61.2 Worked problems on Fourie rseries of non-periodic functions over a range of 2ˇ
62. Even and odd functions and half-range Fourier series
62.1. Even and odd functions
62.2. Fourier cosine and Fourier sine series
62.3. Half-range Fourier series
63. Fourier series over any range
63.1. Expansion of a periodic function of period L
63.2. Half-range Fourier series for functions defined over range L
64. A numerical method of harmonic analysis
64.1. Introduction
64.2. Harmonic analysis on data given in tabular or graphical form
64.3. Complex waveform considerations
65. The complex or exponential form of a Fourier series
65.1. Introduction
65.2. Exponential or complex notation
65.3. Complex coefficients
65.4. Symmetry relationships
65.5. The frequency spectrum
65.6. Phasors
Section L: Z-Transforms
66. An introduction to z-transforms
66.1. Sequences
66.2. Some properties of z-transforms
66.3. Inverse z-transforms
66.4. Using z-transforms to solve difference equations
Revision Test 17
Section M: Statistics and probability
67. Presentation of statistical data
67.1. Some statistical terminology
67.2. Presentation of ungrouped data
67.3. Presentation of grouped data
68. Mean, median, mode and standard deviation
68.1. Measures of central tendency
68.2. Mean, median and mode for discrete data
68.3. Mean, median and mode for grouped data
68.4. Standard deviation
68.5. Quartiles, deciles and percentiles
69. Probability
69.1. Introduction to probability
69.2. Laws of probability
69.3. Worked problems on probability
69.4. Further worked problems on probability
69.5. Permutations and combinations
69.6. Bayes’ theorem
Revision Test 18
70. The binomial and Poisson distributions
70.1. The binomial distribution
70.2. The Poisson distribution
71. The normal distribution
71.1. Introduction to the normal distribution
71.2. Testing for a normal distribution
72. Linear correlation
72.1. Introduction to linear correlation
72.2. The Pearson product-moment formula for determining the linear correlation coefficient
72.3. The significance of a coefficient of correlation
72.4. Worked problems on linear correlation
73. Linear regression
73.1. Introduction to linear regression
73.2. The least-squares regression lines
73.3. Worked problems on linear regression
Revision Test 19
74. Sampling and estimation theories
74.1. Introduction
74.2. Sampling distributions
74.3. The sampling distribution of the means
74.4. The estimation of population parameters based on a large sample size
74.5. Estimating the mean of a population based on a small sample size
75. Significance testing
75.1. Hypotheses
75.2. Type I and type II errors
75.3. Significance tests for population means
75.4. Comparing two sample means
76. Chi-square and distribution-free tests
76.1. Chi-square values
76.2. Fitting data to theoretical distributions
76.3. Introduction to distribution-free tests
76.4. The sign test
76.5. Wilcoxon signed-rank test
76.6. The Mann–Whitney test
Revision Test 20
Essential formulae
Answers to Practice Exercises
Index