Page 79 - Basic Structured Grid Generation

68 Basic Structured Grid Generation

Further important equations may be obtained by considering the consistency of
different representations of Gauss’s formula through the equations
2
2
∂ ∂ r ∂ ∂ r
= ,
α
α
∂u γ ∂u ∂u β ∂u β ∂u ∂u γ
or
∂ ∂a α ∂ ∂a α
= ,
∂u γ ∂u β ∂u β ∂u γ
which gives
∂ µ ∂ µ
a
0 = ( a µ + b αβ N) − ( αγ µ + b αγ N)
αβ
∂u γ ∂u β
µ µ
∂ αβ ∂ αγ µ ∂a µ ∂a µ
µ
= − a µ + αβ − αγ
∂u γ ∂u β ∂u γ ∂u β
∂N ∂N ∂b αβ ∂b αγ
+b αβ γ − b αγ β + γ − β N
∂u ∂u ∂u ∂u
µ µ
∂ αβ ∂ αγ µ µ
δ
δ
= − + − αγ δβ − b αβ b γδ a δµ + b αγ b βδ a δµ a µ
αβ δγ
∂u γ ∂u β

µ µ ∂b αβ ∂b αγ
+ b µγ − αγ µβ + − N,
b
αβ γ β
∂u ∂u
using both eqns (3.121) and (3.122). Since a 1 , a 2 ,and N are an independent set of
vectors, it follows that we must have both
µ µ
∂
αβ ∂ αγ δ µ δ µ δµ δµ
− + − αγ δβ − b αβ b γδ a + b αγ b βδ a = 0 (3.123)
αβ δγ
∂u γ ∂u β
and
µ µ ∂b αβ ∂b αγ
b µγ − b − = 0 (3.124)
αβ αγ µβ + γ β
∂u ∂u
for all possible values of the free (unrepeated) indices.
We can appeal to eqn (3.64) to re-write eqn (3.123) as
µ δµ δµ
R = b αγ b βδ a − b αβ b γδ a (3.125)
.αβγ
and multiplication through by a νµ (with summation over µ), using eqn (3.65), gives
R ναβγ = b αγ b βν − b αβ b γν . (3.126)
Since we know that the curvature tensor R ναβγ has only one independent component
R 1212 , eqns (3.126) reduce to a single equation
2
R 1212 = b 11 b 22 − (b 12 ) = det(b αβ ), (3.127)
and this is Gauss’s equation. It has the consequence that since R 1212 is an intrin-
sic quantity, derivable entirely in terms of the covariant metric tensor a αβ ,then