Page 84 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 73

Exercise 17. Making use of eqn (3.26), show that if A is a surface vector with covari-
˜
ant components given by the 2×1 column vector A (with elements A α ) and background
cartesian components given by the 3 × 1 column vector A (with elements A i ), then
√
˜
A = aC −1 A (3.150)
or, equivalently,
α
A α = A i ∂y i /∂u . (3.151)
Exercise 18. Show by direct matrix multiplication using eqns (3.143) and (3.147) that

1 T T

C √ J = I 3 − NN , (3.152)
a
where here N stands for the column vector of cartesian components of the surface
1 T
normal vector and I 3 is the unit 3 × 3 matrix, and hence that √ J is not a right
a
inverse for C.
1
2
Now consider a surface vector field V(u ,u ), defined at all points of the surface and
having the property that it is everywhere tangential to the surface. Then the divergence
of V is given by an expression analogous to eqn (1.134):
∂V ∂V ∂V
α 1 2
∇· V = a · = a · + a ·
∂u α ∂u 1 ∂u 2
1 ∂V ∂V
= √ (a 2 × N) · + (N × a 1 ) · (3.153)
a ∂u 1 ∂u 2
1 ∂V i
= √ C iα α (3.154)
a ∂u
in terms of the background cartesian components V i of V. (Here the index i is summed
from 1 to 3, while α is summed from 1 to 2.) These expressions are non-conservative.
A conservative form follows by using eqn (3.144):

1 ∂ ∂
∇· V = √ ((a 2 × N) · V) + ((N × a 1 ) · V) − 2κ m N · V,
a ∂u 1 ∂u 2
but now the last term vanishes because V is a tangential vector. Hence we have
1 ∂ ∂
∇· V = √ ((a 2 × N) · V) + ((N × a 1 ) · V) (3.155)
a ∂u 1 ∂u 2
1 ∂
= √ (C iα V i ). (3.156)
a ∂u α
1
2
For any surface scalar field ϕ(u ,u ), the gradient ∇ϕ must be a surface vector
field, according to eqn (3.139). Thus we can combine the results of eqns (3.141) and
2
(3.156) to give a formula for the Laplacian ∇ ϕ =∇ · ∇ϕ:
1 ∂ 1 ∂ϕ
2
∇ ϕ = √ √ C iα C iβ . (3.157)
a ∂u α a ∂u β