3rd Edition

# Vector Analysis and Cartesian Tensors, Third edition

By P C Kendall, D.E. Bourne Copyright 1992
316 Pages
by CRC Press

316 Pages
by CRC Press

304 Pages
by Routledge

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
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
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