Page 24 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 13

i
The Kronecker symbol δ has corresponding matrix elements given by the 3 × 3
j
identity matrix I. It may be interpreted as a second-order mixed tensor, where which-
ever of the covariant or contravariant components occurs first is immaterial, since if we
substitute T = I in either of the transformation rules (1.81) or (1.83) we obtain T = I
i
in view of eqn (1.60). Thus δ is a mixed tensor which has the same components on any
j
co-ordinate system. The corresponding linear operator is just the identity operator I,
which for any vector u satisfies
i j i j i
Iu = (δ g i ⊗ g )u = δ g i u = g i u = u.
j j
The following representations of I may then be deduced:
i j ij j j
I = g ij g ⊗ g = g g i ⊗ g j = g j ⊗ g = g ⊗ g j . (1.87)
i
ij
Thus g ij , g ,and δ are associated tensors.
j
Covariant, contravariant, and mixed tensors of higher order than two may be defined
in terms of transformation rules following the pattern in eqns (1.76), (1.78), (1.80), and
(1.82), though it may not be convenient to express these rules in matrix terms. For
example, covariant and contravariant third-order tensors U ijk and U ijk respectively
must follow the transformation rules:
j
l
i
m
∂x ∂x ∂x n ijk ∂x ∂x ∂x k lmn
U ijk = U lmn , U = U . (1.88)
j
i
m
l
∂x ∂x ∂x k ∂x ∂x ∂x n
The alternating symbol e ijk defined by

 1 if (i, j, k) is an even permutation of (1, 2, 3)

e ijk = e ijk = −1if (i, j, k) is an odd permutation of (1, 2, 3) (1.89)

0 otherwise

is not a (generalized) third-order tensor. Applying the left-hand transformation of
eqns (1.88) gives, using the properties of determinants and eqns (1.61) and (1.67),

m
l
∂x ∂x ∂x n −1 g
e lmn = (det B)e ijk = J e ijk = e ijk . (1.90)
j
i
∂x ∂x ∂x k g
Similarly we obtain
i
j
∂x ∂x ∂x k lmn ijk ijk g ijk
e = (det A)e = Je = e . (1.91)
l
m
∂x ∂x ∂x n g
It follows that third-order covariant and contravariant tensors respectively are
defined by
√
ε ijk = ge ijk (1.92)
and
1 ijk
ijk
ε = √ e . (1.93)
g
√
Applying the appropriate transformation law to ε ijk now gives ge lmn = ε lmn ,
ijk
as required, and similarly for ε . These tensors, known as the alternating tensors,