Page 23 - Basic Structured Grid Generation

12 Basic Structured Grid Generation

or, equivalently,
T
T = ATA . (1.79)
.j j
We can also define mixed second-order tensors T and T , for which the transfor-
i .i
mation rules are
k
.j ∂x ∂x j .l
T i = T k (1.80)
i
∂x ∂x l
T
T = BTA , (1.81)
and
i
i ∂x ∂x l k
T ..j = T .l (1.82)
k
∂x ∂x j
T
T = ATB . (1.83)
Exercise 6. Show from the transformation rules (1.80) and (1.82) that the quantities
k
T and T .k are invariants.
.k k
Given two vectors u and v, second-order tensors can be generated by taking products
of covariant or contravariant vector components, giving the covariant tensor u i v j ,the
i j
i
j
contravariant tensors u v , and the mixed tensors u v j and u i v . In this case these
tensors are said to be associated, since they are all derived from an entity which
can be written in absolute, co-ordinate-free, terms, as u ⊗ v; this is called the dyadic
product of u and v. The dyadic product may also be regarded as a linear operator
which acts on vectors w according to the rule
(u ⊗ v)w = u(v · w), (1.84)

an equation which has various co-ordinate representations, such as
j
j
(u i v j )w = u i (v j w ).
It may also be expressed in the following various ways:
i j i j i j j i
u ⊗ v = u i v j g ⊗ g = u v g i ⊗ g j = u v j g i ⊗ g = u i v g ⊗ g j (1.85)
with summation over i and j in each case.
.j
i
ij
In general, covariant, contravariant, and mixed components T ij ,T ,T ,T ,are
i .j
associated if there exists an entity T, a linear operator which can operate on vectors,
such that
j
i
ij
g ⊗ g j = T g i ⊗ g .
T = T ij g ⊗ g = T g i ⊗ g j = T ..j i i j (1.86)
i ..j
Thus the action of T on a vector u could be represented typically by
i
j
j i
i j
j
i
Tu = (T ij g ⊗ g )u = T ij g (g · u) = T ij g u = T ij u g = v,
where v has covariant components
j
v i = T ij u .