Understanding Engineering Mathematics

Cover

Contents

Preface

To the Student

1 Number and Arithmetic

Prerequisites

Objectives

Motivation

A note about calculators

1.1 Review

1.1.1 Types of numbers

1.1.2 Use of inequality signs

1.1.3 Highest common factor and lowest common multiple

1.1.4 Manipulation of numbers

1.1.5 Handling fractions

1.1.6 Factorial and combinatorial notation – permutations and combinations

1.1.7 Powers and indices

1.1.8 Decimal notation

1.1.9 Estimation

1.2 Revision

1.2.1 Types of numbers

1.2.2 Use of inequality signs

1.2.3 Highest common factor and lowest common multiple

1.2.4 Manipulation of numbers

1.2.5 Handling fractions

An electrical example – resistances in parallel

1.2.6 Factorial and combinatorial notation – permutations and combinations

1.2.7 Powers and indices

Examples

1.2.8 Decimal notation

Examples

1.2.9 Estimation

1.3 Reinforcement

1.3.1 Types of numbers

1.3.2 Use of inequality signs

1.3.3 Highest common factor and lowest common multiple

1.3.4 Manipulation of numbers

1.3.5 Handling fractions

1.3.6 Factorial and combinatorial notation – permutations and

combinations

1.3.7 Powers and indices

1.3.8 Decimal notation

1.3.9 Estimation

1.4 Applications

Answers to reinforcement exercises

1.3.1 Types of numbers

1.3.2 Use of inequality signs

1.3.3 Highest common factor and lowest common multiple

1.3.4 Manipulation of numbers

1.3.5 Handling fractions

1.3.6 Factorial and combinatorial notation

1.3.7 Powers and indices

1.3.8 Decimal notation

1.3.9 Estimation

2 Algebra

Prerequisites

Objectives

Motivation

2.1 Review

2.1.1 Multiplication of linear expressions

2.1.2 Polynomials

2.1.3 Factorisation of polynomials by inspection

2.1.4 Simultaneous equations

2.1.5 Equalities and identities

2.1.6 Roots and factors of a polynomial

2.1.7 Rational functions

2.1.8 Algebra of rational functions

2.1.9 Division and the remainder theorem

2.1.10 Partial fractions

2.1.11 Properties of quadratic expressions and equations

2.1.12 Powers and indices for algebraic expressions

2.1.13 The binomial theorem

2.2 Revision

2.2.1 Multiplication of linear expressions

Examples

2.2.2 Polynomials

Examples

2.2.3 Factorisation of polynomials by inspection

Example

2.2.4 Simultaneous equations

Example

2.2.5 Equalities and identities

Example

2.2.6 Roots and factors of a polynomial

Example

2.2.7 Rational functions

Examples

2.2.8 Algebra of rational functions

Example

Examples

Example

Examples

2.2.9 Division and the remainder theorem

2.2.10 Partial fractions

2.2.11 Properties of quadratic expressions and equations

Example

a positive

a

0/

a negative

a

0/

Example

2.2.12 Powers and indices for algebraic expressions

2.2.13 Binomial theorem

Example

2.3 Reinforcement

2.3.1 Multiplication of linear expressions

2.3.2 Polynomials

2.3.3 Factorisation of polynomials by inspection

2.3.4 Simultaneous equations

2.3.5 Equalities and identities

2.3.6 Roots and factors of a polynomial

2.3.7 Rational functions

2.3.8 Algebra of rational functions

2.3.9 Division and the remainder theorem

2.3.10 Partial fractions

2.3.11 Properties of quadratic expressions and equations

2.3.12 Powers and indices for algebraic expressions

2.3.13 The binomial theorem

2.4 Applications

Answers to reinforcement exercises

2.3.1 Multiplication of linear expressions

2.3.2 Polynomials

2.3.3 Factorisation of polynomials by inspection

2.3.4 Simultaneous equations

2.3.5 Equalities and identities

2.3.6 Roots and factors of a polynomial

2.3.7 Rational functions

2.3.8 Algebra of rational functions

2.3.9 Division and the remainder theorem

2.3.10 Partial fractions

2.3.11 Properties of quadratic expressions and equations

2.3.12 Powers and indices for algebraic expressions

2.3.13 The binomial theorem

3 Functions and Series

Prerequisites

Objectives

Motivation

A note about rigour

3.1 Review

3.1.1 Definition of a function

3.1.2 Plotting the graph of a function

3.1.3 Formulae

3.1.4 Odd and even functions

3.1.5 Composition of functions

3.1.6 Inequalities

3.1.7 Inverse of a function

3.1.8 Series and sigma notation

3.1.9 Finite series

3.1.10 Infinite series

3.1.11 Infinite binomial series

3.2 Revision

3.2.1 Definition of a function

Example

Examples

3.2.2 Plotting the graph of a function

3.2.3 Formulae

3.2.4 Odd and even functions

3.2.5 Composition of functions

Example

3.2.6 Inequalities

Example

3.2.7 Inverse of a function

Example

3.2.8 Series and sigma notation

Example

3.2.9 Finite series

Example

3.2.10 Infinite series

Example

3.2.11 Infinite binomial series

Example

3.3 Reinforcement

3.3.1 Definition of a function

3.3.2 Plotting the graph of a function

3.3.3 Formulae

3.3.4 Odd and even functions

3.3.5 Composition of functions

3.3.6 Inequalities

3.3.7 Inverse of a function

3.3.8 Series and sigma notation

3.3.9 Finite series

3.3.10 Infinite series

3.3.11 Infinite binomial series

3.4 Applications

Answers to reinforcement exercises

3.3.1 Definition of a function

3.3.2 Plotting the graph of a function

3.3.3 Formulae

3.3.4 Odd and even functions

3.3.5 Composition of functions

3.3.6 Inequalities

3.3.7 Inverse of a function

3.3.8 Series and sigma notation

3.3.9 Finite series

3.3.10 Infinite series

3.3.11 Infinite binomial series

4 Exponential and Logarithm Functions

Prerequisites

Objectives

Motivation

4.1 Review

4.1.1 y

an, n= an integer

4.1.2 The general exponential function ax

4.1.3 The natural exponential function ex

4.1.4 Manipulation of the exponential function

4.1.5 Logarithms to general base

4.1.6 Manipulation of logarithms

4.1.7 Some applications of logarithms

4.2 Revise

4.2.1 y

an, n= an integer

4.2.2 The general exponential function ax

4.2.3 The natural exponential function ex

Problem 1

Problem 2

Problem 3

Problem 4

4.2.4 Manipulation of the exponential function

4.2.5 Logarithms to general base

4.2.6 Manipulation of logarithms

4.2.7 Some applications of logarithms

4.3 Reinforcement

4.3.1 y

an, n= an integer

4.3.2 The general exponential function ax

4.3.3 The natural exponential function ex

4.3.4 Manipulation of the exponential function

4.3.5 Logarithms to general base

4.3.6 Manipulation of logarithms

4.3.7 Some applications of logarithms

4.4 Applications

Answers to reinforcement exercises

4.3.1 y

an, n= an integer

4.3.2 The general exponential function ax

4.3.3 The natural exponential function ex

4.3.4 Manipulation of the exponential function

4.3.5 Logarithms to general base

4.3.6 Manipulation of logarithms

4.3.7 Some applications of logarithms

5 Geometry of Lines, Triangles and Circles

Prerequisites

Objectives

Motivation

5.1 Review

5.1.1 Division of a line in a given ratio

5.1.2 Intersecting and parallel lines and angular measurement

5.1.3 Triangles and their elementary properties

5.1.4 Congruent triangles

5.1.5 Similar triangles

5.1.6 The intercept theorem

5.1.7 The angle bisector theorem

5.1.8 Pythagoras’ theorem

5.1.9 Lines and angles in a circle

5.1.10 Cyclic quadrilaterals

5.2 Revision

5.2.1 Division of a line in a given ratio

5.2.2 Intersecting and parallel lines and angular measurement

5.2.3 Triangles and their elementary properties

5.2.4 Congruent triangles

5.2.5 Similar triangles

5.2.6 The intercept theorem

5.2.7 The angle bisector theorem

5.2.8 Pythagoras’ theorem

5.2.9 Lines and angles in a circle

5.2.10 Cyclic quadrilaterals

5.3 Reinforcement

5.3.1 Division of a line in a given ratio

5.3.2 Intersecting and parallel lines and angular measurement

5.3.3 Triangles and their elementary properties

5.3.4 Congruent triangles

5.3.5 Similar triangles

5.3.6 The intercept theorem

5.3.7 The angle bisector theorem

5.3.8 Pythagoras’ theorem

5.3.9 Lines and angles in a circle

5.3.10 Cyclic quadrilaterals

5.4 Applications

Answers to reinforcement exercises

5.3.1 Division of a line in a given ratio

5.3.2 Intersecting and parallel lines and angular measurement

5.3.3 Triangles and their elementary properties

5.3.4 Congruent triangles

5.3.5 Similar triangles

5.3.6 The intercept theorem

5.3.7 The angle bisector theorem

5.3.8 Pythagoras’ theorem

5.3.9 Lines and angles in a circle

5.3.10 Cyclic quadrilaterals

6 Trigonometry

Prerequisites

Objectives

Motivation

6.1 Review

6.1.1 Radian measure and the circle

6.1.2 Definition of the trig ratios

6.1.3 Sine and cosine rules and solutions of triangles

6.1.4 Graphs of the trigonometric functions

6.1.5 Inverse trigonometric functions

6.1.6 The Pythagorean identities Ò cos2

1

6.1.7 Compound angle formulae

6.1.8 Trigonometric equations

6.1.9 The acos

bsin

form

6.2 Revision

6.2.1 Radian measure and the circle

6.2.2 Definition of the trig ratios

6.2.3 Sine and cosine rules and solutions of triangles

6.2.4 Graphs of the trigonometric functions

6.2.5 Inverse trigonometric functions

6.2.6 The Pythagorean identities Ò cos2

1

6.2.7 Compound angle formulae

6.2.8 Trigonometric equations

6.2.9 The acos

bsin

form

6.3 Reinforcement

6.3.1 Radian measure and the circle

6.3.2 Definitions of the trig ratios

6.3.3 Sine and cosine rules and the solution of triangles

6.3.4 Graphs of trigonometric functions

6.3.5 Inverse trigonometric functions

6.3.6 The Pythagorean identities Ò cos2

1

6.3.7 Compound angle formulae

6.3.8 Trigonometric equations

6.3.9 The acos

bsin

form

6.4 Applications

Answers to reinforcement exercises

6.3.1 Radian measure and the circle

6.3.2 Definitions of the trig ratios

6.3.3 Sine and cosine rules and the solution of triangles

6.3.4 Graphs of trigonometric functions

6.3.5 Inverse trigonometric functions

6.3.6 The Pythagorean identities Ò cos2

1

6.3.7 Compound angle formulae

6.3.8 Trigonometric equations

6.3.9 The acos

bsin

form

7 Coordinate Geometry

Prerequisites

Objectives

Motivation

7.1 Review

7.1.1 Coordinate systems in a plane

7.1.2 Distance between two points

7.1.3 Midpoint and gradient of a line

7.1.4 Equation of a straight line

7.1.5 Parallel and perpendicular lines

7.1.6 Intersecting lines

7.1.7 Equation of a circle

7.1.8 Parametric representation of curves

7.2 Revision

7.2.1 Coordinate systems in a plane

Example

7.2.2 Distance between two points

7.2.3 Midpoint and gradient of a line

7.2.4 Equation of a straight line

7.2.5 Parallel and perpendicular lines

7.2.6 Intersecting lines

Example

7.2.7 Equation of a circle

Problem

7.2.8 Parametric representation of curves

7.3 Reinforcement

7.3.1 Coordinate systems in a plane

7.3.2 Distance between two points

7.3.3 Midpoint and gradient of a line

7.3.4 Equation of a straight line

7.3.5 Parallel and perpendicular lines

7.3.6 Intersecting lines

7.3.7 Equation of a circle

7.3.8 Parametric representation of curves

7.4 Applications

Answers to reinforcement exercises

7.3.1 Coordinate systems in a plane

7.3.2 Distance between two points

7.3.3 Midpoint and gradient of a line

7.3.4 Equation of a straight line

7.3.5 Parallel and perpendicular lines

7.3.6 Intersecting lines

7.3.7 Equation of a

circle

7.3.8 Parametric representation of curves

8 Techniques of Differentiation

Prerequisites

Objectives

Motivation

8.1 Review

8.1.1 Geometrical interpretation of differentiation

8.1.2 Differentiation from first principles

8.1.3 Standard derivatives

8.1.4 Rules of differentiation

8.1.5 Implicit differentiation

8.1.6 Parametric differentiation

8.1.7 Higher order derivatives

8.2 Revision

8.2.1 Geometrical interpretation of differentiation

8.2.2 Differentiation from first principles

8.2.3 Standard derivatives

8.2.4 Rules of differentiation

8.2.5 Implicit differentiation

Example

8.2.6 Parametric differentiation

8.2.7 Higher order derivatives

8.3 Reinforcement

8.3.1 Geometrical interpretation of differentiation

8.3.2 Differentiation from first principles

8.3.3 Standard derivatives

8.3.4 Rules of differentiation

8.3.5 Implicit differentiation

8.3.6 Parametric differentiation

8.3.7 Higher order derivatives

8.4 Applications

Answers to reinforcement exercises

8.3.1 Geometrical interpretation of differentiation

8.3.2 Differentiation from first principles

8.3.3 Standard derivatives

8.3.4 Rules of differentiation

8.3.5 Implicit differentiation

8.3.6 Parametric differentiation

8.3.7 Higher order derivatives

9 Techniques of Integration

Prerequisites

Objectives

Motivation

9.1 Review

9.1.1 Definition of integration

9.1.2 Standard integrals

9.1.3 Addition of integrals

9.1.4 Simplifying the integrand

9.1.5 Linear substitution in integration

9.1.6 The du= f

x/ dx substitution

9.1.7 Integrating rational functions

9.1.8 Using trig identities in integration

9.1.9 Using trig substitutions in integration

9.1.10 Integration by parts

9.1.11 Choice of integration methods

9.1.12 The definite integral

9.2 Revision

9.2.1 Definition of integration

9.2.2 Standard integrals

9.2.3 Addition of integrals

9.2.4 Simplifying the integrand

9.2.5 Linear substitution in integration

9.2.6 The du= f

x/ dx substitution

9.2.7 Integrating rational functions

Example

9.2.8 Using trig identities in integration

9.2.9 Using trig substitutions in integration

9.2.10 Integration by parts

9.2.11 Choice of integration methods

9.2.12 The definite integral

9.3 Reinforcement

9.3.1 Definition of integration

9.3.2 Standard integrals

9.3.3 Addition of integrals

9.3.4 Simplifying the integrand

9.3.5 Linear substitution in integration

9.3.6 The du= f

x/ dx substitution

9.3.7 Integrating rational functions

9.3.8 Using trig identities in integration

9.3.9 Using trig substitutions in integration

9.3.10 Integration by parts

9.3.11 Choice of integration methods

9.3.12 The definite integral

9.4 Applications

Answers to reinforcement exercises

9.3.1 Definition of integration

9.3.2 Standard integrals

9.3.3 Addition of integrals

9.3.4 Simplifying the integrand

9.3.5 Linear substitution in integration

9.3.6 The du= f

x/ dx substitution

9.3.7 Integrating rational functions

9.3.8 Using trig

identities

in integration

9.3.9 Using trig substitutions in integration

9.3.10

Integration

by parts

9.3.11 Choice of integration methods

9.3.12 The definite integral

10 Applications of Differentiation and Integration

Prerequisites

Objectives

Motivation

10.1 Review

10.1.1 The derivative as a gradient and rate of change

10.1.2 Tangent and normal to a curve

10.1.3 Stationary points and points of inflection

10.1.4 Curve sketching in Cartesian coordinates

10.1.5 Applications of integration Ò area under a curve

10.1.6 Volume of a solid of revolution

10.2 Revise

10.2.1 The derivative as a gradient and rate of change

10.2.2 Tangent and normal to a curve

10.2.3 Stationary points and points of inflection

10.2.4 Curve sketching in Cartesian coordinates

10.2.5 Applications of integration Ò area under a curve

10.2.6 Volume of a solid of revolution

10.3 Reinforcement

10.3.1 The derivative as a gradient and rate of change

10.3.2 Tangent and normal to a curve

10.3.3 Stationary points and points of inflection

10.3.4 Curve sketching in Cartesian coordinates

10.3.5 Applications of integration Ò area under a curve

10.3.6 Volume of a solid of revolution

10.4 Applications

Answers to reinforcement exercises

10.3.1 The derivative as a gradient and rate of change

10.3.2 Tangent and normal to a curve

10.3.3 Stationary points and points of inflection

10.3.4 Curve sketching in Cartesian coordinates

10.3.5 Applications of integration Ò area under a curve

10.3.6 Volume of a solid of revolution

11 Vectors

Prerequisites

Objectives

Motivation

11.1 Introduction – representation of a vector quantity

Exercise on 11.1

Answers

11.2 Vectors as arrows

Exercise on 11.2

Answers

11.3 Addition and subtraction of vectors

Exercises on 11.3

Answers

11.4 Rectangular Cartesian coordinates in three dimensions

Exercise on 11.4

Answer

11.5 Distance in Cartesian coordinates

Problem 11.1

Exercise on 11.5

Answer

11.6 Direction cosines and ratios

Problem 11.2

Exercise on 11.6

Answer

11.7 Angle between two lines through the origin

Exercise on 11.7

Answer

11.8 Basis vectors

Problem 11.3

Exercises on 11.8

Answers

11.9 Properties of vectors

Problem 11.4

Problem 11.5

Problem 11.6

Exercises on 11.9

Answers

11.10 The scalar product of two vectors

Problem 11.7

Problem 11.8

Exercises on 11.10

Answers

11.11 The vector product of two vectors

Problem 11.9

Exercises on 11.11

Answers

11.12 Vector functions

Exercise on 11.12

Answer

11.13 Differentiation of vector functions

Problem 11.10

Problem 11.11

Problem 11.12

Problem 11.13

Exercises on 11.13

11.14 Reinforcement

11.15 Applications

11.16 Answers to reinforcement exercises

12 Complex Numbers

Prerequisites

Objectives

Motivation

12.1 What are complex numbers?

Problem 12.1

Exercise on 12.1

Answer

12.2 The algebra of complex numbers

Problem 12.2

Problem 12.3

Problem 12.4

Problem 12.5

Problem 12.6

Exercise on 12.2

Answer

12.3 Complex variables and the Argand plane

Exercises on 12.3

Answers

12.4 Multiplication in polar form

Problem 12.7

Problem 12.8

Exercises on 12.4

Answers

12.5 Division in polar form

Problem 12.9

Exercise on 12.5

Answer

12.6 Exponential form of a complex number

Problem 12.10

Problem 12.11

Exercises on 12.6

12.7 De Moivre’s theorem for integer powers

Problem 12.12

Exercise on 12.7

Answer

12.8 De Moivre’s theorem for fractional powers

Problem 12.13

Exercises on 12.8

Answer

12.9 Reinforcement

12.10 Applications

12.11 Answers to reinforcement exercises

13 Matrices and Determinants

Prerequisites

Objectives

Motivation

13.1 An overview of matrices and determinants

13.2 Definition of a matrix and its elements

Problem 13.1

Problem 13.2

Exercise on 13.2

Answer

13.3 Adding and multiplying matrices

Problem 13.3

Problem 13.4

Problem 13.5

Problem 13.6

Problem 13.7

Exercises on 13.3

Answers

13.4 Determinants

Problem 13.8

Problem 13.9

Problem 13.10

Exercises on 13.4

Answers

13.5 CramerÌs rule for solving a system of linear equations

Problem 13.11

Exercise on 13.5

Answer

13.6 The inverse matrix

Problem 13.12

Problem 13.13

Exercise on 13.6

Answer

13.7 Eigenvalues and eigenvectors

Problem 13.14

Exercise on 13.7

Answer

13.8 Reinforcement

13.9 Applications

13.10 Answers to reinforcement exercises

14 Analysis for Engineers – Limits, Sequences, Iteration, Series and All That

Prerequisites

Objectives

Motivation

14.1 Continuity and irrational numbers

Problem 14.1

Exercises on 14.1

Answers

14.2 Limits

Problem 14.2

Problem 14.3

Problem 14.4

Problem 14.5

Exercises on 14.2

Answers

14.3 Some important limits

Exercise on 14.3

Answer

14.4 Continuity

Problem 14.6

Exercise on 14.4

Answer

14.5 The slope of a curve

Problem 14.7

Exercise on 14.5

Answer

14.6 Introduction to infinite series

Exercise on 14.6

14.7 Infinite sequences

Problem 14.8

Problem 14.9

Exercise on 14.7

Answer

14.8 Iteration

Exercise on 14.8

Answer

14.9 Infinite series

Problem 14.10

Problem 14.11

Exercise on 14.9

Answer

14.10 Tests for convergence

Problem 14.12

Problem 14.13

Problem 14.14

Exercise on 14.10

Answer

14.11 Infinite power series

Problem 14.15

Problem 14.16

Exercise on 14.11

Answer

14.12 Reinforcement

14.13 Applications

14.14 Answers to reinforcement exercises

15 Ordinary Differential Equations

Prerequisites

Objectives

Motivation

15.1 Introduction

Problem 15.1

Exercise on 15.1

Answer

15.2 Definitions

Problem 15.2

Problem 15.3

Problem 15.4

Problem 15.5

Exercises on 15.2

Answers

15.3 First order equations Ò direct integration and separation of variables

Problem 15.6

Problem 15.7

Problem 15.8

Exercise on 15.3

Answer

15.4 Linear equations and integrating factors

Problem 15.9

Problem 15.10

Problem 15.11

Problem 15.12

Problem 15.13

Exercise on 15.4

Answer

15.5 Second order linear homogeneous differential equations

Problem 15.14

Problem 15.15

Problem 15.16

Exercises on 15.5

Answers

15.6 The inhomogeneous equation

Problem 15.17

Problem 15.18

Problem 15.19

Problem 15.20

Problem 15.21

Problem 15.22

Exercises on 15.6

Answers

15.7 Reinforcement

15.8 Applications

15.9 Answers to reinforcement exercises

16 Functions of More than One Variable – Partial Differentiation

Prerequisites

Objectives

Motivation

16.1 Introduction

Exercises on 16.1

Answers

16.2 Function of two variables

Exercise 16.2

Answer

16.3 Partial differentiation

Problem 16.1

Exercise on 16.3

Answer

16.4 Higher order derivatives

Problem 16.2

Exercises on 16.4

Answers

16.5 The total differential

Problem 16.3

Problem 16.4

Problem 16.5

Problem 16.6

Exercises on 16.5

Answers

16.6 Reinforcement

16.7 Applications

16.8 Answers to reinforcement exercises

17 An Appreciation of Transform Methods

17.1 Introduction

Prerequisites

Objectives

Motivation

17.2 The Laplace transform

Problem 17.1

Exercises on 17.2

Answers

17.3 Laplace transforms of the elementary functions

Problem 17.2

Problem 17.3

Problem 17.4

Exercises on 17.3

Answers

17.4 Properties of the Laplace transform

1. The Laplace transformis linear

2. The first shift theorem

3. The Laplace transformof the derivative

Problem 17.5

Problem 17.6

Exercises on 17.4

Answers

17.5 The inverse Laplace transform

Problem 17.7

Exercise on 17.5

Answer

17.6 Solution of initial value problems by Laplace transform

Problem 17.8

Problem 17.9

Exercises on 17.6

Answers

17.7 Linear systems and the principle of superposition

Exercise on 17.7

Answer

17.8 Orthogonality relations for trigonometric functions

Exercise on 17.8

17.9 The Fourier series expansion

Exercise on 17.9

Answer

17.10 The Fourier coefficients

Problem 17.10

Exercise on 17.10

17.11 Reinforcement

17.12 Applications

17.13 Answers to reinforcement exercises

Index