Vector Analysis and Cartesian Tensors

Preface

Chapter 1 Rectangular Cartesian Coordinates and Rotation of Axes


1.1 Rectangular Cartesian Coordinates


1.2 Direction Cosines and Direction Ratios


1.3 Angles Between Lines through the Origin


1.4 The Orthogonal Projection of One Line on Another


1.5 Rotation of Axes


1.6 The Summation Convention and its Use


1.7 Invariance with Respect to a Rotation of the Axes


1.8 Matrix Notation

Chapter 2 Scalar and Vector Algebra


2.1 Scalars


2.2 Vectors: Basic Notions


2.3 Multiplication of a Vector by a Scalar


2.4 Addition and Subtraction of Vectors


2.5 The Unit Vectors i, j, k


2.6 Scalar Products


2.7 Vector Products


2.8 The Triple Scalar Product


2.9 The Triple Vector Product


2.10 Products of Four Vectors


2.11 Bound Vectors

Chapter 3 Vector Functions of a Real Variable. Differential Geometry of Curves


3.1 Vector Functions and their Geometrical Representation


3.2 Differentiation of Vectors


3.3 Differentiation Rules


3.4 The Tangent to a Curve. Smooth, Piecewise Smooth, and Simple Curves


3.5 Arc Length


3.6 Curvature and Torsion


3.7 Applications in Kinematics

Chapter 4 Scalar and Vector Fields


4.1 Regions


4.2 Functions of Several Variables


4.3 Definitions of Scalar and Fields


4.4 Gradient of a Scalar Field


4.5 Properties of Gradient


4.6 The Divergence and Curl of a Vector Field


4.7 The Del-Operator


4.8 Scalar Invariant Operators


4.9 Useful Identities


4.10 Cylindrical and Spherical Polar Coordinates


4.11 General Orthogonal Curvilinear Coordinates


4.12 Vector Components in Orthogonal Curvilinear Coordinates


4.13 Expressions for Grad Ω, Div F, Curl F, and ∇2 in Orthogonal Curvilinear Coordinates


4.14 Vector Analysis in n-Dimensional Space

Chapter 5 Line, Surface, and Volume Integrals


5.1 Line Integral of a Scalar Field


5.2 Line Integrals of a Vector Field


5.3 Repeated Integrals


5.4 Double and Triple Integrals


5.5 Surfaces


5.6 Surface Integrals


5.7 Volume Integrals

Chapter 6 Integral Theorems


6.1 Introduction


6.2 The Divergence Theorem (Gauss's Theorem)


6.3 Green's Theorems


6.4 Stokes's Theorem


6.5 Limit Definitions of Div F and Curl F


6.6 Geometrical and Physical Significance of Divergence and Curl

Chapter 7 Applications in Potential Theory


7.1 Connectivity


7.2 The Scalar Potential


7.3 The Vector Potential


7.4 Poisson's Equation


7.5 Poisson's Equation in Vector Form


7.6 Helmholtz's Theorem


7.7 Solid Angles

Chapter 8 Cartesian Tensors


8.1 Introduction


8.2 Cartesian Tensors: Basic Algebra


8.3 Isotropic Tensors


8.4 Tensor Fields


8.5 The Divergence Theorem in Tensor Field Theory

Chapter 9 Representation Theorems for Isotropic Tensor Functions


9.1 Introduction


9.2 Diagonalization of Second Order Symmetrical Tensors


9.3 Invariants of Second Order Symmetrical Tensors


9.4 Representation of Isotropic Vector Functions


9.5 Isotropic Scalar Functions of Symmetrical Second Order Tensors


9.6 Representation of an Isotropic Tensor Function

Appendix 1 Determinants

Appendix 2 The Chain Rule for Jacobians

Appendix 3 Expressions for Grad, Div, Curl, and ∇2 in Cylindrical and Spherical Polar Coordinates

Answers to Exercises

Index