From Calculus to Chaos An Introduction to

What is calculus really for? This book is a highly readable introduction to applications of calculus, from Newton's time to the present day. These often involve questions of dynamics, i.e. of how - and why - things change with time. Problems of this kind lie at the heart of much of appliedmathematics, physics, and engineering. From Calculus to Chaos takes a fresh approach to the subject as a whole, by moving from first steps to the frontiers, and by highlighting only the most important and interesting ideas, which can get lost amid a snowstorm of detail in conventional texts. Thebook is aimed at a wide readership, and assumes only some knowledge of elementary calculus. There are exercises (with full solutions) and simple but powerful computer programs which are suitable even for readers with no previous computing experience. David Acheson's book will inspire new studentsby providing a foretaste of more advanced mathematics and showing just how interesting the subject can be.

Introduction

1

(10)

The beginning of dynamics

1

(4)

From calculus to chaos

5

(5)

A brief review of calculus

10

(11)

Introduction

10

(1)

Some elementary results

11

(2)

Taylor series

13

(3)

The functions ex and log x

16

(5)

Exercises

19

(2)

Ordinary differential equations

21

(16)

Introduction

21

(2)

First-order linear equations

23

(2)

First-order nonlinear equations

25

(4)

Second-order linear equations

29

(2)

Second-order nonlinear equations

31

(2)

Phase space

33

(4)

Exercises

35

(2)

Computer solution methods

37

(17)

Introduction

37

(1)

Euler's method

38

(2)

Computer implementation of Euler's method

40

(5)

Systems of differential equations

45

(3)

More accurate step-by-step methods

48

(6)

Exercises

52

(2)

Elementary oscillations

54

(17)

Introduction

54

(1)

The linear oscillator

55

(4)

Multiple modes of oscillation

59

(2)

Coupled oscillators

61

(3)

Nonlinear oscillations

64

(7)

Exercises

69

(2)

Planetary motion

71

(22)

Introduction

71

(3)

Equations of motion under a central force

74

(2)

The equal-area rule

76

(1)

Differential equation for the orbit

77

(1)

Orbits under an inverse-square law

78

(3)

A numerical approach

81

(3)

The two-body problem

84

(1)

The three-body problem

85

(8)

Exercises

91

(2)

Waves and diffusion

93

(14)

Introduction

93

(1)

Wave motion

94

(5)

The diffusion equation

99

(8)

Exercises

105

(2)

The best of all possible worlds?

107

(12)

Introduction

107

(1)

The concept of `action'

108

(2)

The calculus of variations

110

(3)

Lagrange's equations of motion

113

(6)

Exercises

117

(2)

Fluid flow

119

(14)

Introduction

119

(1)

The geometry of fluid motion

120

(2)

The equations of viscous flow

122

(3)

Very viscous flow

125

(2)

The case of small viscosity

127

(6)

Exercises

131

(2)

Instability and catastrophe

133

(19)

Introduction

133

(3)

Linear stability theory

136

(2)

Multiple solutions; bifurcation

138

(2)

Sudden changes of state

140

(2)

Imperfection and catastrophe

142

(4)

Instability of motion

146

(6)

Exercises

150

(2)

Nonlinear oscillations and chaos

152

(16)

Introduction

152

(1)

Limit cycles; the van der Pol equation

153

(3)

Conditions for chaos

156

(2)

The Lorenz equations

158

(3)

Chaotic mixing: stretch and fold

161

(2)

One route to chaos: period-doubling

163

(1)

Multiple solutions and `jumps'

163

(5)

Exercises

165

(3)

The not-so-simple pendulum

168

(19)

Introduction

168

(2)

Pendulums from the past

170

(3)

A vibrated pendulum

173

(3)

Chaotic pendulums

176

(2)

Not quite the Indian Rope Trick

178

(9)

Exercises

184

(3)

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From Calculus to Chaos An Introduction to Dynamics

Summary

Table of Contents

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