Page 38 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 27

It follows that R ijkl is skew-symmetric in the first two indices as well as the last
two, that is
R ijkl =−R ijlk

and
R ijkl =−R jikl .

Furthermore
R ijkl = R klij .

Hence R ijkl in any curved three-dimensional space has only six independent com-
ponents, namely R 1212 , R 2323 , R 1313 , R 1213 , R 1223 ,and R 2313 .
In a Euclidean space, by eqns (1.179) and (1.180) all components of R ijkl are zero,
and so, according to eqn (1.177), second-order covariant derivatives of arbitrary covari-
ant vectors u i satisfy
u i,jk = u i,kj .
It can be shown that a similar commutative property applies to second-order covariant
derivatives of covariant or contravariant tensors of any order in a Euclidean space.

1.10 Orthogonal curvilinear co-ordinates

i
A curvilinear co-ordinate system {x }={ξ, η, ς} is orthogonal if the covariant base
vectors at any point are mutually orthogonal. It follows that the contravariant base vec-
tors are parallel to their respective covariant base vectors and also mutually orthogonal.
So we have
g 12 = g 23 = g 13 = 0 (1.182)
and
g 12 = g 23 = g 13 = 0.
An example mentioned above is a spherical polar co-ordinate system with metric
tensor given by eqn (1.19).
It is convenient to put
√ √ √
g 11 = h 1 , g 22 = h 2 , g 33 = h 3 , (1.183)
with

g 11 = 1/h 1 , g 22 = 1/h 2 , g 33 = 1/h 3 ,
where h 1 ,h 2 ,h 3 are called scale factors.Then g = g 11 g 22 g 33 and
√
g = h 1 h 2 h 3 . (1.184)
Using eqns (1.108) and (1.100) to evaluate Christoffel symbols, we get the useful
expressions
∂h i i 1 ∂h i
[ii, i]= h i i , = i ,(no summation over i) (1.185)
ii
∂x h i ∂x