Page 63 - Basic Structured Grid Generation

52 Basic Structured Grid Generation

1
1 ∂a 12 ∂a 22 ∂a 22
22 = a 22 2 − − a 12 (3.52)
2a ∂v ∂u ∂v
1
2 ∂a 22 ∂a 22 ∂a 12
22 = a 11 + a 12 − 2
2a ∂v ∂u ∂v
1
2 ∂a 12 ∂a 11 ∂a 11
11 = a 11 2 − − a 12
2a ∂u ∂v ∂u
1
2 2 ∂a 22 ∂a 11
12 = 21 = a 11 − a 12 .
2a ∂u ∂v
There should be no difficulty in practice in distinguishing between the surface
Christoffel symbols and their three-dimensional version in eqn (1.99). The use of either
Greek or Roman indices will indicate the context.

Exercise 7. For the surface of revolution given by eqn (3.32), show that the Christoffel
symbols are

[11, 1]= f f + g g , [12, 2]=[21, 2]= ff

[22, 1]= −ff , [11, 2]=[12, 1]=[21, 1]=[22, 2]= 0 (3.53)
and
2 −1

f
1 = (f 2 + g ) (f f + g g ), 2 = 2 = f −1
11 12 21
2 −1

1 =−(f 2 + g ) ff , 1 = 1 = 2 = 2 = 0. (3.54)
22 12 21 11 22
Exercise 8. For the spherical surface defined by eqn (3.4), with u = θ, v = φ, show
that the only non-vanishing Christoffel symbols are
1
2
[12, 2]=[21, 2]=−[22, 1]=− 22 = sin θ cos θ, 12 = 2 21 = cot θ.
It is straightforward to obtain the results analogous to eqns (1.106) and (1.107),
namely
∂a αβ δ δ
=[γα, β]+ [γβ, α]= a δβ γα + a δα γβ (3.55)
∂u γ
and
∂a αβ δα β δβ α
=−a δγ − a . (3.56)
δγ
∂u γ
Surface covariant differentiation and intrinsic surface derivatives along a surface
curve can now be defined. Following eqns (1.120), (1.122), and (2.31), a surface vector
A, for example, has surface covariant derivatives
∂A α ∂A α α γ ∂A ∂A α γ
α
A = · a = + A , A α,β = · a α = (3.57)
,β β β γβ β β − A γ
αβ
∂u ∂u ∂u ∂u
and intrinsic derivatives
δA α α du β α β δA α du β β
= A ,β = A λ , = A α,β = A α,β λ (3.58)
,β
δs ds δs ds
along a surface curve with unit tangent vector λ.