Appendix A: Geometric Quantities
161
a 1,1 = a 1,2 = a 1,3 = a 2,3 =
⎧
⎨
⎩
0
0
0
⎫
⎬
⎭
, a 2,2 =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−R cos
Θ 2
−R sin
Θ 2
0
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
, a 3,2 =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
− sin
Θ 2
cos
Θ 2
0
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(A.22)
The covariant and mixed components of the curvature tensor are
b αβ = a α,β · a 3 =
0 0
0 −R
, b
β
α = a
βγ
· b αγ =
0 0
0 −
1
R
.
(A.23)
The components of the shifter tensor are
μ
β
α = δ
β
α − Θ
3 b
β
α =
⎡
⎣
1
0
0 1 +
Θ
3
R
⎤
⎦ .
(A.24)
The Christoffel symbols of the second kind for the point at the mid-surface are
Γ
1
αβ = Γ
2
αβ =
0 0
0 0
.
(A.25)
Therefore, the covariant derivatives and the abbreviations
n
ϕ αβ can be obtained as
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n
v 1|1 =
n
v 1,1
n
v 1|2 =
n
v 1,2
n
v 2|1 =
n
v 2,1
n
v 2|2 =
n
v 2,2
and
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n
ϕ 11 =
n
v 1,1
n
ϕ 12 =
n
v 1,2
n
ϕ 21 =
n
v 2,1
n
ϕ 22 =
n
v 2,2 + R
n
v 3
n
ϕ 31 =
n
v 3,1
n
ϕ 32 =
n
v 3,2 −
1
R
n
v 2
(A.26)
A.3 Spherical Structure
The Cartesian coordinate system (X
1 , X
2 , X
3 ) and the curvilinear coordinate system
(Θ
1 , Θ
2 , Θ
3 ) of a spherical structure are shown in Fig. A.3.
Here, R denotes the radius of the mid-surface, and r is the radius of an arbitrary
large spherical surface. The curvilinear coordinates are defined as
Θ
1
= Rβ, Θ
2
= α, Θ
3
= r − R.
(A.27)