Page 61 - Basic Structured Grid Generation

50 Basic Structured Grid Generation

1
2
1
2
Under transformations from surface co-ordinates (u ,u ) to (u , u ) we then have
γ
γ
γ
∂u ∂u δ αβ ∂u ∂u δ ∂u ∂u δ γδ
e = − = Je ,
α
2
1
∂u ∂u β ∂u ∂u 2 ∂u ∂u 1
and, similarly,
α
∂u ∂u β −1
e αβ = J e γδ ,
γ δ
∂u ∂u
where J is the Jacobian of the transformation:
1
∂u ∂u 1

∂u 1 ∂u 2

J = .
∂u 2 ∂u 2

1 2
∂u ∂u
Thus e αβ and e αβ transform like relative tensors. Derivations similar to those result-
ing in the definitions (1.92) and (1.93) show that absolute (surface) tensors are given
by ε αβ and ε αβ ,where
1 αβ √
αβ
ε = √ e , ε αβ = ae αβ . (3.44)
a
Note that
γ
ε αβ ε αγ = e αβ e αγ = δ . (3.45)
β
Exercise 6. Show that
αβ γδ
ε ε a αγ a βδ = ε αβ ε γδ a αγ βδ = 2.
a
We can now say, for example, that, given two unit surface vectors λ and µ,a
β
α
consistent tensor expression is given by ε αβ λ µ . Comparing this with eqn (1.94), we
see that it represents the component of the vector product of λ and µ in the direction of
a unit vector normal to the surface, a direction which remains invariant under changes
of surface co-ordinates. Thus, if θ is the angle between the directions λ, µ (in the sense
of a positive right-handed screw rotation from λ to µ), we have
α
β
sin θ = ε αβ λ µ . (3.46)
The condition for the orthogonality of two surface directions λ, µ may now be
written either, using eqn (3.39), as
α β
a αβ λ µ = 0, (3.47)
or, from eqn (3.46), as
α
β
ε αβ λ µ =±1. (3.48)
In this section it may be seen how the metrical properties of a surface, i.e. the
measurement of lengths of curves on the surface, angles between intersecting curves,
and areas, may be derived with reference to the first fundamental form of the surface,
and in particular to the covariant metric tensor a αβ .Since a αβ itself was derived by
using the properties of the enveloping Euclidean space, we may say that the metrical