Page 20 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 9

Hence
i i
u = u · g , (1.48)
and, similarly,
u i = u · g . (1.49)
i
A similar procedure shows, incidentally, that
j
g i = g ij g , (1.50)
and
ij
i
g = g g j . (1.51)
We then easily deduce that
i ij
u = g u j (1.52)
and
j
u i = g ij u . (1.53)
These equations may be interpreted as demonstrating that the action of g ij on u j
j
and that of g ij on u are effectively equivalent to ‘raising the index’ and ‘lowering the
index’, respectively.
It is straightforward to show that the scalar product of vectors u and v is given by
i i i j ij
u · v = u v i = u i v = g ij u v = g u i v j (1.54)
and hence that the magnitude of a vector u is given by

i j ij
|u|= g ij u u = g u i u j . (1.55)
It is important to note the special transformation properties of covariant and con-
travariant components under a change of curvilinear co-ordinate system. We consider
i
another system of co-ordinates x , i = 1, 2, 3, related to the first system by the trans-
formation equations
3
2
1
i
i
x = x (x ,x ,x ), i = 1, 2, 3. (1.56)
These equations are assumed to be invertible and differentiable. In particular, dif-
ferentials in the two systems are related by the chain rule
∂x i
i j
dx = dx , (1.57)
∂x j
or,inmatrixterms,
 1   1 
dx dx
 dx 2  = A  dx 2  , (1.58)
dx 3 dx 3
where we assume that the matrix A of the transformation, with i-j element equal to
i
j
∂x /∂x , has a determinant not equal to zero, so that eqn (1.58) may be inverted. We
define the Jacobian J of the transformation as
J = det A. (1.59)