Page 60 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 49

A surface vector A has covariant and contravariant components with respect to the
surface base vectors given, respectively, by

α
α
A α = A · a α , A = A · a . (3.35)
Since eqn (3.16) can be written
α
du du β
1 = a αβ , (3.36)
ds ds
α
α
it follows by comparison with eqn (3.26) that du /ds = λ represents the contravariant
3
components of a unit surface vector. Viewed from E this vector λ has cartesian
components
dy i ∂y i du α ∂y i α
λ i = = = λ (3.37)
ds ∂u α ds ∂u α
(which may be regarded as a set of direction cosines) and background contravariant
curvilinear components
i
dx i ∂x du α ∂x i
i α
λ = = = λ . (3.38)
ds ∂u α ds ∂u α
The angle θ between directions specified by unit surface vectors λ and µ each
α
α β
β
satisfying a αβ λ λ = 1and a αβ µ µ = 1isgiven by
cos θ = λ · µ = λ i µ i
in cartesian components, or, by eqn (3.37),
∂y i ∂y i α β α β
cos θ = λ µ = a αβ λ µ (3.39)
α
∂u ∂u β
using eqn (3.17). This result may be compared with the general equations for a scalar
product in eqn (1.54).
1
2
Unit surface vectors λ, µ tangential to the u and u co-ordinate curves at a point
must have contravariant components given by, respectively,
1 1
1 2 1 2
λ = √ , λ = 0 and µ = 0, µ = √ . (3.40)
a 11 a 22
According to eqn (3.39) the angle θ between the co-ordinate curves is given by

1 2 a 12
cos θ = a 12 λ µ = √ . (3.41)
a 11 a 22
The co-ordinate curves form an orthogonal network if a 12 = F = 0 everywhere.
The element of surface area dσ given by the parallelogram with sides formed by
2
1
the line-segments a 1 du , a 2 du tangential to the co-ordinate curves at a point is
√ 1 2
dσ = a du du , (3.42)
analogously to eqn (1.44).
It is sometimes convenient to have at our disposal the two-dimensional alternating
symbol e αβ = e αβ satisfying
11 22 12 21
e = e 11 = e = e 22 = 0, e = e 12 = 1, e = e 21 =−1. (3.43)