Page 29 - Basic Structured Grid Generation

18 Basic Structured Grid Generation

j
Differentiating a vector field u with respect to x gives
∂u ∂ i ∂u i i ∂g i ∂u i i k ∂u i i k
= (u g i ) = g i + u = g i + u g k = + u g i ,
kj
ij
∂x j ∂x j ∂x j ∂x j ∂x j ∂x j
j
with some re-arrangement of indices. Thus ∂u/∂x (itself a vector field) is given by
∂u i
= u g i , (1.119)
,j
∂x j
where
∂u i
i i k
u = + u (1.120)
,j j kj
∂x
i
is called the covariant derivative of the contravariant vector u .
A similar calculation gives
∂u i
= u i,j g , (1.121)
∂x j
where the covariant derivative of the covariant vector u i is given by
∂u i k
u i,j = − u k . (1.122)
ij
∂x j
Exercise 11. Using the definitions (1.120) and (1.122) and the transformation rules
(1.70), (1.72), and (1.109), show that u i and u i,j satisfy the transformation rules for
,j
mixed and covariant second-order tensors, respectively.
These tensors are associated, since the equations
∂u i i ik
= u g i = u i,j g = u i,j g g k
,j
∂x j
imply, comparing coefficients, that
ik
i
u = g u k,j , (1.123)
,j
after some re-arrangement of indices; or, more simply, since
i
j
i
j
u g i ⊗ g = u i,j g ⊗ g .
,j
Clearly we also have
∂u ∂u
i i
u = · g and u i,j = · g i . (1.124)
,j j j
∂x ∂x
Covariant differentiation can also be applied to tensor fields. With a second-order
tensor T as given in eqn (1.86), we give the following example:
∂T ∂ j j ∂T ij i j ∂g i j i ∂g j
= (T ij g ⊗ g ) = g ⊗ g + T ij ⊗ g + T ij g ⊗
∂x k ∂x k ∂x k ∂x k ∂x k
∂T ij i j i l j i j l
= g ⊗ g − T ij g ⊗ g − T ij g ⊗ g
kl
kl
∂x k

∂T ij l l i j
= − T lj − T il g ⊗ g ,
kj
ki
∂x k