164
Appendix A: Geometric Quantities
a i j = a i · a j =
⎡
⎢
⎢
⎣
1
0
0
0 R
2 sin
2
Θ
1
R
0
0
0
1
⎤
⎥
⎥
⎦ , a
i j
= a
−1
i j =
⎡
⎢
⎢
⎢
⎢
⎣
1
0
0
0
1
R 2 sin
2
Θ
1
R
0
0
0
1
⎤
⎥
⎥
⎥
⎥
⎦
.
(A.34)
The contravariant base vectors at the mid-surface are
a 1 =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
cos
Θ 1
R
cos
Θ 2
cos
Θ 1
R
sin
Θ 2
− sin
Θ 1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, a 2 =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−
sin
Θ 2
sin
Θ 1
R
cos
Θ 2
sin
Θ 1
R
0
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, a 3 =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sin
Θ 1
R
cos
Θ 2
sin
Θ 1
R
sin
Θ 2
cos
Θ 1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
.
(A.35)
The partial derivatives of the covariant base vectors at the mid-surface are
a 1,1 = −
1
R
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sin
Θ
1
R
cos
Θ
2
sin
Θ
1
R
sin
Θ
2
cos
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
a 1,2 = a 2,1 =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
− cos
Θ
1
R
sin
Θ
2
cos
Θ
1
R
cos
Θ
2
0
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
,
a 2,2 =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
− sin
Θ
1
R
cos
Θ
2
− sin
Θ
1
R
sin
Θ
2
0
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
,
a 3,1 =
1
R
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
cos
Θ
1
R
cos
Θ
2
cos
Θ
1
R
sin
Θ
2
− sin
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
a 3,2 =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
− sin
Θ
1
R
sin
Θ
2
sin
Θ
1
R
cos
Θ
2
0
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
,
a 1,3 = a 2,3 = a 3,3 =
⎧
⎨
⎩
0
0
0
⎫
⎬
⎭
.
(A.36)
The covariant and mixed components of the curvature tensor are