Page 25 - Basic Structured Grid Generation

14 Basic Structured Grid Generation

are required, for example, when forming correct vector expressions in curvilinear co-
ordinate systems.
In particular, the vector product of two vectors u and v is given by
ijk
j k i
u × v = ε u j v k g i = ε ijk u v g , (1.94)
with summation over i, j, k. The component forms of the scalar triple product of
vectors u, v, w are
ijk
k
i j
u · (v × w) = ε u i v j w k = ε ijk u v w . (1.95)
The alternating symbols themselves may be called relative (rather than absolute) ten-
sors, which means that when the tensor transformation law is applied as in eqns (1.90)
and (1.91) a power of J (the weight of the relative tensor) appears on the right-hand
side. Thus according to (1.90) e ijk is a relative tensor of weight −1, while according
to eqn (1.91) e ijk (although it takes exactly the same values as e ijk ) is a relative tensor
of weight 1.

1.6 Christoffel symbols and covariant differentiation

In curvilinear co-ordinates the base vectors will generally vary in magnitude and direc-
tion from one point to another, and this causes special problems for the differentiation
of vector and tensor fields. In general, differentiation of covariant base vectors eqn (1.4)
j
with respect to x satisfies
2
2
∂ r ∂ r
∂g i ∂g j
= = = . (1.96)
j
i
∂x j ∂x ∂x i ∂x ∂x j ∂x i
Expressing the resulting vector (for a particular choice of i and j)asalinearcom-
bination of base vectors gives
∂g i k k
=[ij, k]g = g k , (1.97)
ij
∂x j
k
with summation over k. The coefficients [ij, k], in eqn (1.97) are called Christoffel
ij
symbols of the first and second kinds, respectively. Taking appropriate scalar products
on eqn (1.97) gives
∂g i
[ij, k]= · g k (1.98)
∂x j
and
k ∂g i k
= · g . (1.99)
ij j
∂x
Both [ij, k] and k are symmetric in i and j by eqn (1.96). We also have, by
ij
eqn (1.51),
k ∂g i kl kl
= · (g g l ) = g [ij, l] (1.100)
ij j
∂x
with summation over l. Similarly,
l
[ij, k]= g kl . (1.101)
ij