Page 22 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 11

or,inmatrixform,
   
u 1 u 1
 u 2  = B  u 2  . (1.71)
u 3 u 3
∂ϕ
The set of components (where ϕ is a scalar field) found in eqn (1.13) can be said
∂x j
to constitute a covariant vector, since by the usual chain rule they transform according
to eqn (1.70), i.e.
j
∂ϕ ∂x ∂ϕ
= .
i
∂x i ∂x ∂x j
Exercise 5. Show that the transformation rule for contravariant components of a
vector is
∂x i
i j
u = u , (1.72)
∂x j
or
 1   1 
u u
 u 2  = A  u 2  . (1.73)
u 3 u 3

Note the important consequence that the scalar product (1.54) is an invariant quantity
(a true scalar), since it is unaffected by co-ordinate transformations. In fact

i
k i k
∂x ∂x ∂x ∂x
i j j k j j
u v i = u v k = u v k = δ u v k = u v j .
j
j
∂x j ∂x i ∂x ∂x i
From eqns (1.64), (1.65), and (1.66), we obtain the transformation rules:
T T T T
(g ) = L L = BLL B = B(g ij )B , (1.74)
ij
T
ij
T
T
ij
T
(g ) = M M = AMM A = A(g )A . (1.75)
In fact g ij is a particular case of a covariant tensor of order two,which maybe
defined here as a set of quantities which take the values T ij , say, when the curvilinear
i
i
co-ordinates x are chosen and the values T ij when a different set x are chosen, with a
transformation rule between the two sets of values being given in co-ordinate form by
k
∂x ∂x l
T ij = i j T kl (1.76)
∂x ∂x
with summation over k and l,orinmatrix form
T
T = BT B . (1.77)
Similarly, g ij is a particular case of a contravariant tensor of order two.This is
defined as an entity which has components T ij obeying the transformation rules
i
ij ∂x ∂x j kl
T = T (1.78)
k
∂x ∂x l