Page 30 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 19

after some rearrangement of indices, with the help of eqn (1.105). Thus
∂T i j
= T ij,k g ⊗ g , (1.125)
∂x k
where
∂T ij l l
T ij,k = − T lj − T il (1.126)
jk
ik
∂x k
is a covariant tensor of order three. For example, if we put T ij = g ij , it follows, using
eqns (1.16) and (1.102) and substituting into (1.126), that
g ij,k = 0 (1.127)

for all i, j, k. This result follows naturally from the tensor properties of the covariant
derivative and the fact that in cartesian co-ordinate systems covariant derivatives reduce
to straightforward partial derivatives. Since g ij takes constant values in a cartesian
system, the partial derivatives of these values are all zero, and these will transform
to zero under tensor transformation to any other co-ordinate system. It can be shown
similarly that
ij
g = 0 (1.128)
,k
for all i, j, k, where the covariant derivative of general contravariant components T ij
is given by
ij ∂T ij i lj j il
T = + T + T . (1.129)
,k k lk lk
∂x
Covariant derivatives of third-order tensors may also be defined, but it will suffice
here to mention the alternating tensor, which could be written as
i
k
ijk
j
ε g i ⊗ g j ⊗ g k = ε ijk g ⊗ g ⊗ g .
Since both covariant and contravariant components reduce to the array of constants
(1.89) in a cartesian system, a similar argument to that used above for g ij shows that
the covariant derivatives must vanish, i.e.
ijk
ε = 0 (1.130)
,l
and
ε ijk,l = 0 (1.131)
for all i, j, k, l.
It may be shown that the product rule for differentiation is valid for covariant
differentiation; for example,
ij
ij
ij
(T u k ) ,l = T u k,l + T u k .
,l
1.7 Div, grad, and curl

The divergence of a vector field u,where
u = U 1 i 1 + U 2 i 2 + U 3 i 3 , (1.132)