160
Appendix A: Geometric Quantities
The covariant base vectors for an arbitrary point in the shell space are
g 1 =
⎧
⎨
⎩
0
0
−1
⎫
⎬
⎭
, g 2 =
⎧
⎨
⎩
−(R + Θ
3
) sin
Θ
2
(R + Θ
3
) cos
Θ
2
0
⎫
⎬
⎭
, g 3 =
⎧
⎨
⎩
cos
Θ
2
sin
Θ
2
0
⎫
⎬
⎭
. (A.16)
The covariant and contravariant metric tensors in the shell space are
g i j = g i · g j =
⎡
⎣
1
0
0
0
R + Θ
3
2 0
0
0
1
⎤
⎦ , g
i j
= [g i j ]
−1
=
⎡
⎢
⎢
⎣
1
0
0
0
1
R + Θ 3
2 0
0
0
1
⎤
⎥
⎥
⎦ .
(A.17)
The contravariant base vectors in the shell space are
g
1
=
⎧
⎨
⎩
0
0
−1
⎫
⎬
⎭
, g
2
=
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
−
sin
Θ
2
(R + Θ 3 )
cos
Θ
2
(R + Θ 3 )
0
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
, g
3
=
⎧
⎨
⎩
cos
Θ
2
sin
Θ
2
0
⎫
⎬
⎭
.
(A.18)
The covariant base vectors at the mid-surface are
a 1 =
⎧
⎨
⎩
0
0
−1
⎫
⎬
⎭
, a 2 =
⎧
⎨
⎩
−R sin
Θ
2
R cos
Θ
2
0
⎫
⎬
⎭
, a 3 = n =
⎧
⎨
⎩
cos
Θ
2
sin
Θ
2
0
⎫
⎬
⎭
.
(A.19)
The covariant and contravariant metric tensors at the mid-surface are
a i j = a i · a j =
⎡
⎣
1 0 0
0 R
2 0
0 0 1
⎤
⎦ , a
i j
= [a i j ]
−1
=
⎡
⎢
⎣
1 0 0
0
1
R 2 0
0 0 1
⎤
⎥
⎦ .
(A.20)
The contravariant base vectors at the mid-surface are
a
1
=
⎧
⎨
⎩
0
0
−1
⎫
⎬
⎭
, a
2
=
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
−
sin
Θ
2
R
cos
Θ
2
R
0
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
, a
3
= a 3 =
⎧
⎨
⎩
cos
Θ
2
sin
Θ
2
0
⎫
⎬
⎭
.
(A.21)
The partial derivatives of the covariant base vectors at the mid-surface are