# Book:D.E. Bourne/Vector Analysis

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

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

Published $1967$, **Oldbourne**.

### Subject Matter

### 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**