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

    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 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
    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
    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
    Surface integrals
    Volume integrals
    Integral theorems
    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
    The scalar potential
    The vector potential
    Poisson’s equation
    Poisson’s equation in vector form
    Helmholtz’s theorem
    Solid angles
    Cartesian tensors
    Cartesian tensors: basic algebra
    Isotropic tensors
    Tensor fields
    The divergence theorem in tensor field theory
    Representation theorems for isotropic tensor functions
    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


    P C Kendall