Modern Engineering Mathematics

Front Cover
Half Title Page
Title Page
Copyright Page
Contents
Preface
About the authors
Chapter 1 Number, Algebra and Geometry
1.1 Introduction
1.2 Number and arithmetic
1.2.1 Number line
1.2.2 Representation of numbers
1.2.3 Rules of arithmetic
1.2.4 Exercises (1–9)
1.2.5 Inequalities
1.2.6 Modulus and intervals
1.2.7 Exercises (10–14)
1.3 Algebra
1.3.1 Algebraic manipulation
1.3.2 Exercises (15–20)
1.3.3 Equations, inequalities and identities
1.3.4 Exercises (21–32)
1.3.5 Suffïx and sigma notation
1.3.6 Factorial notation and the binomial expansion
1.3.7 Exercises (33–35)
1.4 Geometry
1.4.1 Coordinates
1.4.2 Straight lines
1.4.3 Circles
1.4.4 Exercises (36–43)
1.4.5 Conics
1.4.6 Exercises (44–46)
1.5 Number and accuracy
1.5.1 Rounding, decimal places and significant figures
1.5.2 Estimating the effect of rounding errors
1.5.3 Exercises (47–56)
1.5.4 Computer arithmetic
1.5.5 Exercises (57–59)
1.6 Engineering applications
1.7 Review exercises (1–25)
Chapter 2 Functions
2.1 Introduction
2.2 Basic definitions
2.2.1 Concept of a function
2.2.2 Exercises (1–6)
2.2.3 Inverse functions
2.2.4 Composite functions
2.2.5 Exercises (7–13)
2.2.6 Odd, even and periodic functions
2.2.7 Exercises (14–16)
2.3 Linear and quadratic functions
2.3.1 Linear functions
2.3.2 Least squares fit of a linear function to experimental data
2.3.3 Exercises (17–23)
2.3.4 The quadratic function
2.3.5 Exercises (24–29)
2.4 Polynomial functions
2.4.1 Basic properties
2.4.2 Factorization
2.4.3 Nested multiplication and synthetic division
2.4.4 Roots of polynomial equations
2.4.5 Exercises (30–38)
2.5 Rational functions
2.5.1 Partial fractions
2.5.2 Exercises (39–42)
2.5.3 Asymptotes
2.5.4 Parametric representation
2.5.5 Exercises (43–47)
2.6 Circular functions
2.6.1 Trigonometric ratios
2.6.2 Exercises (48–54)
2.6.3 Circular functions
2.6.4 Trigonometric identities
2.6.5 Amplitude and phase
2.6.6 Exercises (55–66)
2.6.7 Inverse circular (trigonometric) functions
2.6.8 Polar coordinates
2.6.9 Exercises (67–71)
2.7 Exponential, logarithmic and hyperbolic functions
2.7.1 Exponential functions
2.7.2 Logarithmic functions
2.7.3 Exercises (72–80)
2.7.4 Hyperbolic functions
2.7.5 Inverse hyperbolic functions
2.7.6 Exercises (81–88)
2.8 Irrational functions
2.8.1 Algebraic functions
2.8.2 Implicit functions
2.8.3 Piecewise defined functions
2.8.4 Exercises (89–98)
2.9 Numerical evaluation of functions
2.9.1 Tabulated functions and interpolation
2.9.2 Exercises (99–104)
2.10 Engineering application: a design problem
2.11 Engineering application: an optimization problem
2.12 Review exercises (1–23)
Chapter 3 Complex Numbers
3.1 Introduction
3.2 Properties
3.2.1 The Argand diagram
3.2.2 The arithmetic of complex numbers
3.2.3 Complex conjugate
3.2.4 Modulus and argument
3.2.5 Exercises (1–18)
3.2.6 Polar form of a complex number
3.2.7 Euler's formula
3.2.8 Exercises (19–27)
3.2.9 Relationship between circular and hyperbolic functions
3.2.10 Logarithm of a complex number
3.2.11 Exercises (28–33)
3.3 Powers of complex numbers
3.3.1 De Moivre's theorem
3.3.2 Powers of trigonometric functions and multiple angles
3.3.3 Exercises (34–41)
3.4 Loci in the complex plane
3.4.1 Straight lines
3.4.2 Circles
3.4.3 More general loci
3.4.4 Exercises (42–50)
3.5 Functions of a complex variable
3.5.1 Exercises (51–56)
3.6 Engineering application: alternating currents in electrical networks
3.6.1 Exercises (57–58)
3.7 Review exercises (1–34)
Chapter 4 Vector Algebra
4.1 Introduction
4.2 Basic definitions and results
4.2.1 Cartesian coordinates
4.2.2 Scalars and vectors
4.2.3 Addition of vectors
4.2.4 Exercises (1–10)
4.2.5 Cartesian components and basic properties
4.2.6 Complex numbers as vectors
4.2.7 Exercises (11–26)
4.2.8 The scalar product
4.2.9 Exercises (27–40)
4.2.10 The vector product
4.2.11 Exercises (41–56)
4.2.12 Triple products
4.2.13 Exercises (57–65)
4.3 The vector treatment of the geometry of lines and planes
4.3.1 Vector equation of a line
4.3.2 Exercises (66–72)
4.3.3 Vector equation of a plane
4.3.4 Exercises (73–83)
4.4 Engineering application: spin-dryer suspension
4.4.1 Point-particle model
4.5 Engineering application: cable-stayed bridge
4.5.1 A simple stayed bridge
4.6 Review exercises (1–22)
Chapter 5 Matrix Algebra
5.1 Introduction
5.2 Basic concepts, definitions and properties
5.2.1 Definitions
5.2.2 Basic operations of matrices
5.2.3 Exercises (1–11)
5.2.4 Matrix multiplication
5.2.5 Exercises (12–18)
5.2.6 Properties of matrix multiplication
5.2.7 Exercises (19–33)
5.3 Determinants
5.3.1 Exercises (34–50)
5.4 The inverse matrix
5.4.1 Exercises (51–59)
5.5 Linear equations
5.5.1 Exercises (60–71)
5.5.2 The solution of linear equations: elimination methods
5.5.3 Exercises (72–78)
5.5.4 The solution of linear equations: iterative methods
5.5.5 Exercises (79–84)
5.6 Rank
5.6.1 Exercises (85–93)
5.7 The eigenvalue problem
5.7.1 The characteristic equation
5.7.2 Eigenvalues and eigenvectors
5.7.3 Exercises (94–95)
5.7.4 Repeated eigenvalues
5.7.5 Exercises (96–101)
5.7.6 Some useful properties of eigenvalues
5.7.7 Symmetric matrices
5.7.8 Exercises (102–106)
5.8 Engineering application: spring systems
5.8.1 A two-particle system
5.8.2 An n-particle system
5.9 Engineering application: steady heat transfer through composite materials
5.9.1 Introduction
5.9.2 Heat conduction
5.9.3 The three-layer situation
5.9.4 Many-layer situation
5.10 Review exercises (1–26)
Chapter 6 An Introduction to Discrete Mathematics
6.1 Introduction
6.2 Set theory
6.2.1 Definitions and notation
6.2.2 Union and intersection
6.2.3 Exercises (1–8)
6.2.4 Algebra of sets
6.2.5 Exercises (9–17)
6.3 Switching and logic circuits
6.3.1 Switching circuits
6.3.2 Algebra of switching circuits
6.3.3 Exercises (18–29)
6.3.4 Logic circuits
6.3.5 Exercises (30–31)
6.4 Propositional logic and methods of proof
6.4.1 Propositions
6.4.2 Compound propositions
6.4.3 Algebra of statements
6.4.4 Exercises (32–37)
6.4.5 Implications and proofs
6.4.6 Exercises (38–47)
6.5 Engineering application: decision support
6.6 Engineering application: control
6.7 Review exercises (1–23)
Chapter 7 Sequences, Series and Limits
7.1 Introduction
7.2 Sequences and series
7.2.1 Notation
7.2.2 Graphical representation of sequences
7.2.3 Exercises (1–13)
7.3 Finite sequences and series
7.3.1 Arithmetical sequences and series
7.3.2 Geometric sequences and series
7.3.3 Other finite series
7.3.4 Exercises (14–25)
7.4 Recurrence relations
7.4.1 First-order linear recurrence relations with constant coefficients
7.4.2 Exercises (26–28)
7.4.3 Second-order linear recurrence relations with constant coefficients
7.4.4 Exercises (29–35)
7.5 Limit of a sequence
7.5.1 Convergent sequences
7.5.2 Properties of convergent sequences
7.5.3 Computation of limits
7.5.4 Exercises (36–40)
7.6 Infinite series
7.6.1 Convergence of infinite series
7.6.2 Tests for convergence of positive series
7.6.3 The absolute convergence of general series
7.6.4 Exercises (41–49)
7.7 Power series
7.7.1 Convergence of power series
7.7.2 Special power series
7.7.3 Exercises (50–56)
7.8 Functions of a real variable
7.8.1 Limit of a function of a real variable
7.8.2 One-sided limits
7.8.3 Exercises (57–61)
7.9 Continuity of functions of a real variable
7.9.1 Properties of continuous functions
7.9.2 Continuous and discontinuous functions
7.9.3 Numerical location of zeros
7.9.4 Exercises (62–69)
7.10 Engineering application: insulator chain
7.11 Engineering application: approximating functions and Padé approximants
7.12 Review exercises (1–25)
Chapter 8 Differentiation and Integration
8.1 Introduction
8.2 Differentiation
8.2.1 Rates of change
8.2.2 Definition of a derivative
8.2.3 Interpretation as the slope of a tangent
8.2.4 Differentiable functions
8.2.5 Speed, velocity and acceleration
8.2.6 Exercises (1–7)
8.2.7 Mathematical modelling using derivatives
8.2.8 Exercises (8–18)
8.3 Techniques of differentiation
8.3.1 Basic rules of differentiation
8.3.2 Derivative of xr
8.3.3 Differentiation of polynomial functions
8.3.4 Differentiation of rational functions
8.3.5 Exercises (19–25)
8.3.6 Differentiation of composite functions
8.3.7 Differentiation of inverse functions
8.3.8 Exercises (26–33)
8.3.9 Differentiation of circular functions
8.3.10 Extended form of the chain rule
8.3.11 Exercises (34–37)
8.3.12 Differentiation of exponential and related functions
8.3.13 Exercises (38–46)
8.3.14 Parametric and implicit differentiation
8.3.15 Exercises (47–59)
8.4 Higher derivatives
8.4.1 The second derivative
8.4.2 Exercises (60–72)
8.4.3 Curvature of plane curves
8.4.4 Exercises (73–78)
8.5 Applications to optimization problems
8.5.1 Optimal values
8.5.2 Exercises (79–88)
8.6 Numerical differentiation
8.6.1 The chord approximation
8.6.2 Exercises (89–93)
8.7 Integration
8.7.1 Basic ideas and definitions
8.7.2 Mathematical modelling using integration
8.7.3 Exercises (94–102)
8.7.4 Definite and indefinite integrals
8.7.5 The Fundamental Theorem of Calculus
8.7.6 Exercise (103)
8.8 Techniques of integration
8.8.1 Integration as antiderivative
8.8.2 Integration of piecewise-continuous functions
8.8.3 Exercises (10–109)
8.8.4 Integration by parts
8.8.5 Exercises (110–111)
8.8.6 Integration using the general composite rule
8.8.7 Exercises (112–116)
8.8.8 Integration using partial fractions
8.8.9 Exercises (117–118)
8.8.10 Integration involving the circular and hyperbolic functions
8.8.11 Exercises (119–120)
8.8.12 Integration by substitution
8.8.13 Integration involving (ax2+bx+ c)
8.8.14 Exercises (121–126)
8.9 Applications of integration
8.9.1 Volume of a solid of revolution
8.9.2 Centroid of a plane area
8.9.3 Centre of gravity of a solid of revolution
8.9.4 Mean values
8.9.5 Root mean square values
8.9.6 Arclength and surface area
8.9.7 Moments of inertia
8.9.8 Exercises (127–136)
8.10 Numerical evaluation of integrals
8.10.1 The trapezium rule
8.10.2 Simpson's rule
8.10.3 Exercises (137–142)
8.11 Engineering application: design of prismatic channels
8.12 Engineering application: harmonic analysis of periodic functions
8.13 Review exercises (1–39)
Chapter 9 Further Calculus
9.1 Introduction
9.2 Improper integrals
9.2.1 Integrand with an infinite discontinuity
9.2.2 Infinite integrals
9.2.3 Exercise (1)
9.3 Some theorems with applications to numerical methods
9.3.1 Rolle's theorem and the first mean value theorems
9.3.2 Convergence of iterative schemes
9.3.3 Exercises (2–7)
9.4 Taylor's theorem and related results
9.4.1 Taylor polynomials and Taylor's theorem
9.4.2 Taylor and Maclaurin series
9.4.3 L'Hôpital's rule
9.4.4 Exercises (8–20)
9.4.5 Interpolation revisited
9.4.6 Exercises (21–23)
9.4.7 The convergence of iterations revisited
9.4.8 Newton–Raphson procedure
9.4.9 Optimization revisited
9.4.10 Exercises (24–27)
9.4.11 Numerical integration
9.4.12 Exercises (28–31)
9.5 Calculus of vectors
9.5.1 Differentiation and integration of vectors
9.5.2 Exercises (32–36)
9.6 Functions of several variables
9.6.1 Representation of functions of two variables
9.6.2 Partial derivatives
9.6.3 Directional derivatives
9.6.4 Exercises (37–46)
9.6.5 The chain rule
9.6.6 Exercises (47–56)
9.6.7 Successive differentiation
9.6.8 Exercises (57–67)
9.6.9 The total differential and small errors
9.6.10 Exercises (68–75)
9.6.11 Exact differentials
9.6.12 Exercises (76–78)
9.7 Taylor's theorem for functions of two variables
9.7.1 Taylor's theorem
9.7.2 Optimization of unconstrained functions
9.7.3 Exercises (79–87)
9.7.4 Optimization of constrained functions
9.7.5 Exercises (88–93)
9.8 Engineering application: deflection of a built-in column
9.9 Engineering application: streamlines in fluid dynamics
9.10 Review exercises (1–35)
Chapter 10 Introduction to Ordinary Differential Equations
10.1 Introduction
10.2 Engineering examples
10.2.1 The take-off run of an aircraft
10.2.2 Domestic hot-water supply
10.2.3 Hydro-electric power generation
10.2.4 Simple electrical circuits
10.3 The classification of ordinary differential equations
10.3.1 Independent and dependent variables
10.3.2 The order of a differential equation
10.3.3 Linear and nonlinear differential equations
10.3.4 Homogeneous and nonhomogeneous equations
10.3.5 Exercises (1–2)
10.4 Solving differential equations
10.4.1 Solution by inspection
10.4.2 General and particular solutions
10.4.3 Boundary and initial conditions
10.4.4 Analytical and numerical solution
10.4.5 Exercises (3–6)
10.5 First-order ordinary differential equations
10.5.1 A geometrical perspective
10.5.2 Exercises (7–10)
10.5.3 Solution of separable differential equations
10.5.4 Exercises (11–17)
10.5.5 Solution of differential equations of form
10.5.6 Exercises (18–22)
10.5.7 Solution of exact differential equations
10.5.8 Exercises (23–30)
10.5.9 Solution of linear differential equations
10.5.10 Solution of the Bernoulli differential equations
10.5.11 Exercises (31–38)
10.6 Numerical solution of first-order ordinary differential equations
10.6.1 A simple solution method: Euler's method
10.6.2 Analysing Euler's method
10.6.3 Using numerical methods to solve engineering problems
10.6.4 Exercises (39–45)
10.7 Engineering application: analysis of damper performance
10.8 Linear differential equations
10.8.1 Differential operators
10.8.2 Linear differential equations
10.8.3 Exercises (46–54)
10.9 Linear constant-coefficient differential equations
10.9.1 Linear homogeneous constant-coefficient equations
10.9.2 Exercises (55–61)
10.9.3 Linear nonhomogeneous constant-coefficient equations
10.9.4 Exercises (62–65)
10.10 Engineering application: second-order linear constant-coefficient differential equations
10.10.1 Free oscillations of elastic systems
10.10.2 Free oscillations of damped elastic systems
10.10.3 Forced oscillations of elastic systems
10.10.4 Oscillations in electrical circuits
10.10.5 Exercises (66–73)
10.11 Numerical solution of second- and higher-order differential equations
10.11.1 Numerical solution of coupled first-order equations
10.11.2 State-space representation of higher-order systems
10.11.3 Exercises (74–79)
10.12 Qualitative analysis of second-order differential equations
10.12.1 Phase-plane plots
10.12.2 Exercises (80–81)
10.13 Review exercises (1–35)
Chapter 11 Introduction to Laplace Transforms
11.1 Introduction
11.2 The Laplace transform
11.2.1 Definition and notation
11.2.2 Transforms of simple functions
11.2.3 Existence of the Laplace transform
11.2.4 Properties of the Laplace transform
11.2.5 Table of Laplace transforms
11.2.6 Exercises (1–3)
11.2.7 The inverse transform
11.2.8 Evaluation of inverse transforms
11.2.9 Inversion using the first shift theorem
11.2.10 Exercise (4)
11.3 Solution of differential equations
11.3.1 Transforms of derivatives
11.3.2 Transforms of integrals
11.3.3 Ordinary differential equations
11.3.4 Exercise (5)
11.3.5 Simultaneous differential equations
11.3.6 Exercise (6)
11.4 Engineering applications: electrical circuits and mechanical vibrations
11.4.1 Electrical circuits
11.4.2 Mechanical vibrations
11.4.3 Exercises (7–12)
11.5 Review exercises (1–18)
Chapter 12 Introduction to Fourier Series
12.1 Introduction
12.2 Fourier series expansion
12.2.1 Periodic functions
12.2.2 Fourier's theorem
12.2.3 The Fourier coefficients
12.2.4 Functions of period 2
12.2.5 Even and odd functions
12.2.6 Even and odd harmonics
12.2.7 Linearity property
12.2.8 Convergence of the Fourier series
12.2.9 Exercises (1–7)
12.2.10 Functions of period T
12.2.11 Exercises (8–13)
12.3 Functions defined over a finite interval
12.3.1 Full-range series
12.3.2 Half-range cosine and sine series
12.3.3 Exercises (14–23)
12.4 Differentiation and integration of Fourier series
12.4.1 Integration of a Fourier series
12.4.2 Differentiation of a Fourier series
12.4.3 Exercises (24–26)
12.5 Engineering application: analysis of a slider–crank mechanism
12.6 Review exercises (1–21)
Chapter 13 Data Handling and Probability Theory
13.1 Introduction
13.2 The raw material of statistics
13.2.1 Experiments and sampling
13.2.2 Data types
13.2.3 Graphs for qualitative data
13.2.4 Histograms of quantitative data
13.2.5 Alternative types of plot for quantitative data
13.2.6 Exercises (1–5)
13.3 Probabilities of random events
13.3.1 Interpretations of probability
13.3.2 Sample space and events
13.3.3 Axioms of probability
13.3.4 Conditional probability
13.3.5 Independence
13.3.6 Exercises (6–23)
13.4 Random variables
13.4.1 Introduction and definition
13.4.2 Discrete random variables
13.4.3 Continuous random variables
13.4.4 Properties of density and distribution functions
13.4.5 Exercises (24–31)
13.4.6 Measures of location and dispersion
13.4.7 Expected values
13.4.8 Independence of random variables
13.4.9 Scaling and adding random variables
13.4.10 Measures from sample data
13.4.11 Exercises (32–48)
13.5 Important practical distributions
13.5.1 The binomial distribution
13.5.2 The Poisson distribution
13.5.3 The normal distribution
13.5.4 The central limit theorem
13.5.5 Normal approximation to the binomial
13.5.6 Random variables for simulation
13.5.7 Exercises (49–65)
13.6 Engineering application: quality control
13.6.1 Attribute control charts
13.6.2 United States standard attribute charts
13.6.3 Exercises (66–67)
13.7 Engineering application: clustering of rare events
13.7.1 Introduction
13.7.2 Survey of near-misses between aircraft
13.7.3 Exercises (68–69)
13.8 Review exercises (1–13)
Appendix I Tables
Al.1 Some useful results
Al.2 Trigonometric identities
Al.3 Derivatives and integrals
Al.4 Some useful standard integrals
Answers to Exercises
Index
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