Two and Three Dimensional Calculus

Preface xi

1 Revision of One-Dimensional Calculus 1

1.1 Limits and Convergence 1

1.2 Differentiation 3

1.2.1 Rules for Differentiation 5

1.2.2 Mean Value Theorem 7

1.2.3 Taylor’s Series 8

1.2.4 Maxima and Minima 12

1.2.5 Numerical Differentiation 13

1.3 Integration 16

Exercises 22

2 Partial Differentiation 25

2.1 Introduction 25

2.2 Differentials 29

2.2.1 Small Errors 30

2.3 Total Derivative 33

2.4 Chain Rule 36

2.4.1 Leibniz Rule 39

2.4.2 Chain Rule in n Dimensions 41

2.4.3 Implicit Functions 42

2.5 Jacobian 43

2.6 Higher Derivatives 46

2.6.1 Higher Differentials 49

2.7 Taylor’s Theorem 50

2.8 Conjugate Functions 52

2.9 Case Study: Thermodynamics 54

Exercises 58

3 Maxima and Minima 61

3.1 Introduction 61

3.2 Maxima, Minima and Saddle Points 63

3.3 Lagrange Multipliers 74

3.3.1 Generalisations 77

3.4 Optimisation 81

3.4.1 Hill Climbing Techniques 81

Exercises 85

4 Vector Algebra 89

4.1 Introduction 89

4.2 Vector Addition 90

4.3 Components 92

4.4 Scalar Product 94

4.5 Vector Product 97

4.5.1 Scalar Triple Product 102

4.5.2 Vector Triple Product 105

Exercises 106

5 Vector Differentiation 109

5.1 Introduction 109

5.2 Differential Geometry 111

5.2.1 Space Curves 112

5.2.2 Surfaces 120

5.3 Mechanics 129

Exercises 135

6 Gradient, Divergence, and Curl 139

6.1 Introduction 139

6.2 Gradient 139

6.3 Divergence 143

6.4 Curl 145

6.5 Vector Identities 146

6.6 Conjugate Functions 151

Exercises 154

7 Curvilinear Co-ordinates 157

7.1 Introduction 157

7.2 Curved Axes and Scale Factors 157

7.3 Curvilinear Gradient, Divergence, and Curl 161

7.3.1 Gradient 161

7.3.2 Divergence 163

7.3.3 Curl 165

7.4 Further Results and Tensors 166

7.4.1 Tensor Notation 166

7.4.2 Covariance and Contravariance 168

Exercises 171

8 PathIntegrals 173

8.1 Introduction 173

8.2 Integration Along a Curve 173

8.3 Practical Applications 181

Exercises 186

9 Multiple Integrals 191

9.1 Introduction 191

9.2 The Double Integral 191

9.2.1 Rotation and Translation 199

9.2.2 Change of Order of Integration 201

9.2.3 Plane Polar Co-ordinates 203

9.2.4 Applications of Double Integration 208

9.3 Triple Integration 213

9.3.1 Cylindrical and Spherical Polar Co-ordinates 219

9.3.2 Applications of Triple Integration 227

Exercises 233

10 Surface Integrals 241

10.1 Introduction 241

10.2 Green’s Theorem in the Plane 242

10.3 Integration over a Curved Surface 246

10.4 Applications of Surface Integration 253

Exercises 256

11 Integral Theorems 259

11.1 Introduction 259

11.2 Stokes’ Theorem 260

11.3 Gauss’ Divergence Theorem 268

11.3.1 Green’s Second Identity 275

11.4 Co-ordinate-Free Definitions 277

11.5 Applications of Integral Theorems 279

11.5.1 Electromagnetic Theory 279

11.5.1.1 Maxwell’s Equations 279

11.5.2 Fluid Mechanics 283

11.5.3 Elasticity Theory 287

11.5.4 Heat Transfer 297

Exercises 298

12 Solutions and Answers to Exercises 301

References 375

Index 377