Page 12 - Basic Structured Grid Generation

1

Mathematical

preliminaries – vector and tensor

analysis

1.1 Introduction

In this chapter we review the fundamental results of vector and tensor calculus which
form the basis of the mathematics of structured grid generation. We do not feel it
necessary to give derivations of these results from the perspective of modern dif-
ferential geometry; the derivations provided here are intended to be appropriate to
the background of most engineers working in the area of grid generation. Helpful
introductions to tensor calculus may be found in Kay (1988), Kreyzig (1968), and
Spain (1953), as well as many books on continuum mechanics, such as Aris (1962).
Nevertheless, we have tried to make this chapter reasonably self-contained. Some of
the essential results were presented by the authors in Farrashkhalvat and Miles (1990);
this book started at an elementary level, and had the restricted aim, compared with
many of the more wide-ranging books on tensor calculus, of showing how to use
tensor methods to transform partial differential equations of physics and engineer-
ing from one co-ordinate system to another (an aim which remains relevant in the
present context). There are some minor differences in notation between the present
book and Farrashkhalvat and Miles (1990).

1.2 Curvilinear co-ordinate systems and base
vectors in E 3

i
We consider a general set of curvilinear co-ordinates x , i = 1, 2, 3, by which points
3 1 2 3
in a three-dimensional Euclidean space E may be specified. The set {x ,x ,x }
could stand for cylindrical polar co-ordinates {r, θ, z}, spherical polars {r, θ, ϕ},etc.
A special case would be a set of rectangular cartesian co-ordinates, which we shall
generally denote by {y 1 ,y 2 ,y 3 } (where our convention of writing the integer indices
as subscripts instead of superscripts will distinguish cartesian from other systems),
1
3
2
or sometimes by {x, y, z} if this would aid clarity. Instead of {x ,x ,x },it may
occasionally be clearer to use notation such as {ξ, η, ς} without indices.