Cover

Title Page

Copyright

Contents

Preface

Chapter 1 Revision of One‐Dimensional Calculus

1.1 Limits and Convergence

1.2 Differentiation

1.2.1 Rules for Differentiation

1.2.2 Mean Value Theorem

1.2.3 Taylor's Series

1.2.4 Maxima and Minima

1.2.5 Numerical Differentiation

1.3 Integration

Chapter 2 Partial Differentiation

2.1 Introduction

2.2 Differentials

2.2.1 Small Errors

2.3 Total Derivative

2.4 Chain Rule

2.4.1 Leibniz Rule

2.4.2 Chain Rule in n Dimensions

2.4.3 Implicit Functions

2.5 Jacobian

2.6 Higher Derivatives

2.6.1 Higher Differentials

2.7 Taylor's Theorem

2.8 Conjugate Functions

2.9 Case Study: Thermodynamics

Chapter 3 Maxima and Minima

3.1 Introduction

3.2 Maxima, Minima and Saddle Points

3.3 Lagrange Multipliers

3.3.1 Generalisations

3.4 Optimisation

3.4.1 Hill Climbing Techniques

Chapter 4 Vector Algebra

4.1 Introduction

4.2 Vector Addition

4.3 Components

4.4 Scalar Product

4.5 Vector Product

4.5.1 Scalar Triple Product

4.5.2 Vector Triple Product

Chapter 5 Vector Differentiation

5.1 Introduction

5.2 Differential Geometry

5.2.1 Space Curves

5.2.2 Surfaces

5.3 Mechanics

Chapter 6 Gradient, Divergence, and Curl

6.1 Introduction

6.2 Gradient

6.3 Divergence

6.4 Curl

6.5 Vector Identities

6.6 Conjugate Functions

Chapter 7 Curvilinear Co‐ordinates

7.1 Introduction

7.2 Curved Axes and Scale Factors

7.3 Curvilinear Gradient, Divergence, and Curl

7.3.1 Gradient

7.3.2 Divergence

7.3.3 Curl

7.4 Further Results and Tensors

7.4.1 Tensor Notation

7.4.2 Covariance and Contravariance

Chapter 8 Path Integrals

8.1 Introduction

8.2 Integration Along a Curve

8.3 Practical Applications

Chapter 9 Multiple Integrals

9.1 Introduction

9.2 The Double Integral

9.2.1 Rotation and Translation

9.2.2 Change of Order of Integration

9.2.3 Plane Polar Co‐ordinates

9.2.4 Applications of Double Integration

9.3 Triple Integration

9.3.1 Cylindrical and Spherical Polar Co‐ordinates

9.3.2 Applications of Triple Integration

Chapter 10 Surface Integrals

10.1 Introduction

10.2 Green's Theorem in the Plane

10.3 Integration over a Curved Surface

10.4 Applications of Surface Integration

Chapter 11 Integral Theorems

11.1 Introduction

11.2 Stokes' Theorem

11.3 Gauss' Divergence Theorem

11.3.1 Green's Second Identity

11.4 Co‐ordinate‐Free Definitions

11.5 Applications of Integral Theorems

11.5.1 Electromagnetic Theory

11.5.1.1 Maxwell's Equations

11.5.2 Fluid Mechanics

11.5.3 Elasticity Theory

11.5.4 Heat Transfer

Exercises

Chapter 12 Solutions and Answers to Exercises

12.1 Chapter 1

12.2 Chapter 2

12.3 Chapter 3

12.4 Chapter 4

12.5 Chapter 5

12.6 Chapter 6

12.7 Chapter 7

12.8 Chapter 8

12.9 Chapter 9

12.10 Chapter 10

9.11 Chapter 11

References

Index

EULA