Page 40 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 29

taking the scalar product of the gradient vector of the given function with a unit vector
in the required direction.
3
Given a set of curvilinear co-ordinates ξ, η, ς in E , consider the ξ co-ordinate
curve (on which η and ς are constant) at some point P in space. Tangential to the
curve is the covariant base vector g 1 , and a unit vector in this direction is given by
g 1 /|g 1 |. Thus the tangential derivative of the scalar function ϕ at the point P in the ξ-
direction is given by
ξ

∂ϕ g 1
= ·∇ϕ.
∂T |g 1 |
i
Reverting to the notation x for the curvilinear co-ordinates, we can write the tan-
i
gential derivative in the x -direction as
i
x
∂ϕ g i g i j ∂ϕ 1 j ∂ϕ
= ·∇ϕ = · g = δ i ,
∂T |g i | |g i | ∂x j |g i | ∂x j
where we have made use of eqn (1.13), and summation is assumed over j but not i.
So the tangential derivative is, by the usual properties of the Kronecker symbol,
i
x
∂ϕ 1 ∂ϕ 1 ∂ϕ
= = √ (1.192)
∂T |g i | ∂x i g ii ∂x i
with no summation over i.
1
The rate of change of ϕ at a point P in a direction normal to the x co-ordinate
1
surface (the surface on which x is constant) passing through P may be obtained
1
similarly when it is recalled that the contravariant vector g is normal to this surface.
1
Thus we can write the normal derivative of ϕ to the x surface as
1
x 1
∂ϕ g
= ·∇ϕ,
1
∂n |g |
i
so that the normal derivative to the x co-ordinate surface is given, using eqn (1.15), by
i
x i
∂ϕ g j ∂ϕ 1 ij ∂ϕ 1 ij ∂ϕ
= · g = g =
g , (1.193)
i
i
∂n |g | ∂x j |g | ∂x j g ii ∂x j
where summation is assumed over j, but not over i, throughout.