# Vector Analysis and Cartesian Tensors

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Vector Analysis and Cartesian Tensors, Second Edition focuses on the processes, methodologies, and approaches involved in vector analysis and Cartesian tensors, including volume integrals, coordinates, curves, and vector functions. The publication first elaborates on rectangular Cartesian coordinates and rotation of axes, scalar and vector algebra, and differential geometry of curves. Discussions focus on differentiation rules, vector functions and their geometrical representation, scalar and vector products, multiplication of a vector by a scalar, and angles between lines through the origin. The text then elaborates on scalar and vector fields and line, surface, and volume integrals, including surface, volume, and repeated integrals, general orthogonal curvilinear coordinates, and vector components in orthogonal curvilinear coordinates. The manuscript ponders on representation theorems for isotropic tensor functions, Cartesian tensors, applications in potential theory, and integral theorems. Topics include geometrical and physical significance of divergence and curl, Poisson's equation in vector form, isotropic scalar functions of symmetrical second order tensors, and diagonalization of second-order symmetrical tensors. The publication is a valuable reference for mathematicians and researchers interested in vector analysis and Cartesian tensors.

## Table of Contents

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

## Product details

- No. of pages: 266
- Language: English
- Copyright: © Academic Press 1977
- Published: January 1, 1977
- Imprint: Academic Press
- eBook ISBN: 9781483260709

## About the Authors

### D. E. Bourne

### P. C. Kendall

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