Bird's Higher Engineering Mathematics

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