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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

*n*th 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

Subject index

Symbols index

### Biography

C. Evans