3rd Edition

# Vector Analysis and Cartesian Tensors, Third edition

**Also available as eBook on:**

This is a comprehensive and self-contained text suitable for use by undergraduate mathematics, science and engineering students. Vectors are introduced in terms of cartesian components, making the concepts of gradient, divergent and curl particularly simple. The text is supported by copious examples and progress can be checked by completing the many problems at the end of each section. Answers are provided at the back of the book.

Preface

Preface to second edition**1 Rectangular Cartesian coordinates and rotation of axes**Rectangular Cartesian coordinates

Direction cosines and direction ratios

Angles between lines through the origin

The orthogonal projection of one line on another

Rotation of axes

The summation convention and its use

Invariance with respect to a rotation of the axes

Matrix notation

**Scalar and vector algebra**

Scalars

Vectors: basic notions

Multiplication of a vector by a scalar

Addition and subtraction of vectors

The unit vectors i, j, k

Scalar products

Vector products

The triple scalar product

The triple vector product

Products of four vectors

Bound vectors

**Vector functions of a real variable. Differential geometry of curves**

Vector functions and their geometrical representation

Differentiation of vectors

Differentiation rules

The tangent to a curve, Smooth, piecewise smooth and simple curves

Arc length

Curvature and torsion

Applications in kinematics

**Scalar and vector fields**

Regions

Functions of several variables

Definitions of scalar and vector fields

Gradient of a scalar field

Properties of gradient

The divergence and curl of a vector field

The del-operator

Scalar invariant operators

Useful identities

Cylindrical and spherical polar coordinates

General orthogonal curvilinear coordinates

Vector components in orthogonal curvilinear coordinates

Expressions for grad Ω, div F, curl F, and ∆² in orthogonal curvilinear coordinates

Vector analysis in

*n*-dimensional space

Method of steepest Desent

**Line, surface and volume integrals**

Line integral of a scalar field

Line integrals of a vector field

Repeated integrals

Double and triple integrals

Surfaces

Surface integrals

Volume integrals

**Integral theorems**

Introduction

The divergence theorem (Gauss’s Theorem)

Green’s theorems

Stokes’s theorem

Limit definitions of div F and curl F

Geometrical and physical significance of divergence and curl

**Applications in potential theory**

Connectivity

The scalar potential

The vector potential

Poisson’s equation

Poisson’s equation in vector form

Helmholtz’s theorem

Solid angles

**Cartesian tensors**

Introduction

Cartesian tensors: basic algebra

Isotropic tensors

Tensor fields

The divergence theorem in tensor field theory

**Representation theorems for isotropic tensor functions**

Introduction

Diagonalization of second order symmetrical tensors

Invariants of second order symmetrical tensors

Representation of isotropic vector functions

Isotropic scalar functions of symmetrical second order tensors

Representation of an isotropic tensor function

**Appendix A Determinants**

Appendix B Expressions for grand, div, curl, and ∆² in cylindrical and spherical polar coordinates

Appendix C The chain rule for Jacobians

Answers to exercises

Index

Appendix B Expressions for grand, div, curl, and ∆² in cylindrical and spherical polar coordinates

Appendix C The chain rule for Jacobians

Answers to exercises

Index

### Biography

P C Kendall