# Book:D.E. Bourne/Vector Analysis

## D.E. Bourne and P.C. Kendall: Vector Analysis

Published $1967$, Oldbourne.

### 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
4.6 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