Covers all the mathematics required on the first year of a degree or diploma course in engineering.
Preface
Acknowledgements
To the Student
1 Basic Ideas
Arithmetic
Representation
Decimal places and significant figures
Precedence
Set notation
Deductions
Algebra
Rules of elementary algebra
Workshop
Identities and equations
Simultaneous equations
Rational expressions
Rearranging equations
Quadratic equations
Polynomials
The binomial theorem
Workshop
The general binomial theorem
Practical
Summary
Assignment
Further exercises
Further concepts
Indices and logarithms
Inequalities
Rules for inequalities
Partial fractions
Workshop
Set notation
Functions
Methods of proof
Practical
Summary
Exercises
Assignment
Further exercises
Trigonometry and geometry
Coordinate systems
Circular functions
Trigonometrical identities
The form a cos 0 + b sin 0
Solutions of equations
Workshop
Coordinate geometry
The straight line
The circle
The conic sections
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Limits, continuity and differentiation
Limits
The laws of limits
Workshop
Right and left limits
Continuity
Differentiability
Leibniz’s theorem
Techniques of differentiation
Workshop
Logarithmic differentiation
Implicit differentiation
Parametric differentiation
Rates of change
Workshop
Practical
Summary
Exercises
Assignments
Further exercises
Hyperbolic functions
Definitions and identities
Differentiation of hyperbolic functions
Curve sketching
Workshop
Injective functions
Surjective functions
Bijective functions
Pseudo-inverse functions
Differentiation of inverse functions
The inverse circular functions
Workshop
The inverse hypervolic functions
Practical
Summary
Exercises
Assignment
Further exercises
Further differentiation
Tangents and normals
Workshop
Intrinsic coordinates
The catenary
Curvature
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Partial differentiation
Functions
Continuity
Partial derivatives
Higher-order derivatives
Workshop
The formulae for a change of variables
The total differential
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Series expansions and their uses
The mean value property
Taylor’s theorem
Workshop
L’Hospital’s rule
Workshop
Maxima and minima
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Infinite series
Series
Convergence and divergence
Tests for convergence and divergence
Power series
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Complex numbers
Genesis
The complex plane: Argand diagram
Vectorial representation
Further properties of the conjugate
De Moivre’s theorem
Workshop
The nth roots of a complex number
Power series
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Matrices
Notation
Matrix algebra
Workshop
Matrix equations
Zero, identity and inverse matrices
Algebraic rules
Practical
Summary
Exercises
Assignment
Further exercises
Determinants
Notation
Cramer’s rule
Higher-order determinants
Rules for determinants
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Inverse matrices
The inverse of a square matrix
Row transformations
Obtaining inverses
Systematic elimination
Workshop
Pivoting
Practical
Concluding remarks
Summary
Exercises
Assignment
Further exercises
Vectors
Descriptions
Vector addition
Scalar multiplication
Components
The scalar product
Direction ratios and direction cosines
Algebraic properties
Applications
The vector product
Workshop
The triple scalar product
The triple vector product
Differentiation of vectors
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Integration 1
The concept of integration
Rules for integration
Workshop
Integration by inspection
Integration by partial fractions
Integration by parts
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Integration 2
Integration by substitution
Workshop
Special substitutions
Integration using standard forms
Reduction formulae
Workshop
Practical
Summary
Exercises
Assignment
Further ecercises
Integration 3
Definite integration
Improper integrals
Area under the curve
Volume of revolution workshop
Length of a curve
Centres of mass
The theorems of Pappus
Moments of inertia
The perpendicular axis theorem
Average values Radius of gyration
Practical
Summary
Exercises
Assignment
Further exercises
Numerical techniques
The solution of the equation f(x)=0
Graphical methods
Iterative methods
The bisection method
The regula falsi method
The secant method
Newton’s method
Workshop
Approximations to derivatives
Workshop
Numerical integration
The trapezoidal rule
Simpson’s rule
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
First-order differential equations
Terminology
Variables separable equations
Workshop
Linear equations
Workshop
Bernoulli’s equation
Homogeneous equations
Workshop
Reducible equations
Boundary conditions
Practical
Summary
Exercises
Assignment
Further exercises
Second-order differential equations
Linear differential equations
The homogeneous case
Workshop
The non-homogeneous case
The particular solution
Workshop
The breakdown case
Workshop
Higher-order equations
Damping
Resonance
Transient and steady state
Practical
Summary
Exercises
Assignment
Further exercises
Fourier series
Fourier series
Odd and even functions
Sine series and cosine series
Dirichlet’s conditions
Harmonics
Fourier series over any finite interval
Workshop
Further developments
Practical
Summary
Exercises
Assignment
Further exercises
Laplace transforms
Integral transforms
The Heaviside unit function
The Dirac delta function
The inverse Laplace transform
Workshop
Practical
Summary
Exxercises
Assignment
Further exercises
Descriptive statistics
Terminology
Random Samples
Population statistics
Data
Pictorial representations
Pie charts
Bar Charts
Histograms
Grouped data
Cumulative frequency diagrams
Measures of location and measures of spread
The mean
The mode
The median
The range
The mean absolute deviation
The standard deviation
Workshop
Practrical
Summary
Exercises
Assignment
Further exercises
Probability
Concepts
The rules of probability
The sum rule
Mutually exclusive events
Conditional probability
The product rule
Independent events
Complementation rule
A random variable
The analytical method
The relative frequency method
The mathematical method
The binomial distribution
The mean of a probability distribution
Variance of a probability distribution
The Poisson distribution
Approximation for the binomial distribution
Continuous distributions
Mean and variance
The normal distribution
Discrete approximations
Continuity correction
Normal probability paper
Workshop
Practical
Summary
Exercises
Assignment
Further exercises
Hints and solutions
Subject index
Symbols index
