Book:D.E. Bourne/Vector Analysis

Subject Matter

 * Vector Algebra

Contents

 * Preface (July 1966)


 * Chapter 1 Regular 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


 * Chapter 2 Scalar and Vector Algebra


 * 2.1 Scalars
 * 2.2 Vectors: basic notations
 * 2.3 Multiplication of a vector by a scalar
 * 2.4 Addition and subtraction of vectors
 * 2.5 The unit vectors $\mathbf i$, $\mathbf j$, $\mathbf k$
 * 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 vector fields
 * 4.4 Gradient of a scalar field
 * 4.5 Properties of a 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 $\grad \Omega$, $\operatorname {div} \mathbf F$, $\curl \mathbf F$, and $\nabla^2$ in orthogonal curvilinear coordinates


 * 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 $\operatorname {div} \mathbf F$ and $\curl \mathbf F$
 * 6.6 Geometrical and physical significance of divergence and curl


 * Chapter 7 Applications


 * 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


 * Appendix 1 Determinants


 * Appendix 2 The chain rule for Jacobians


 * Appendix 3 Expressions for grad, div, curl, and $\nabla^2$ in cylindrical and spherical polar coordinates


 * Answers to Exercises


 * Index