Appendix A: Geometric Quantities
163
The contravariant metric tensor in the shell space is
g
i j
= [g i j ]
−1
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
R
2
R + Θ 3
2
0
0
0
1
R + Θ 3
2 sin
2
Θ
1
R
0
0
0
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(A.31)
The contravariant base vectors in the shell space are
g
1
=
R
R + Θ 3
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
cos
Θ
1
R
cos
Θ
2
cos
Θ
1
R
sin
Θ
2
− sin
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
g
2
=
1
R + Θ 3
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−
sin
Θ
2
sin
Θ
1
R
cos
Θ
2
sin
Θ
1
R
0
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, g
3
=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sin
Θ
1
R
cos
Θ
2
sin
Θ
1
R
sin
Θ
2
cos
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
.
(A.32)
The covariant 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 = R
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
− sin
Θ
1
R
sin
Θ
2
sin
Θ
1
R
cos
Θ
2
0
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
,
a 3 = n =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sin
Θ
1
R
cos
Θ
2
sin
Θ
1
R
sin
Θ
2
cos
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
.
(A.33)
The covariant and contravariant metric tensors at the mid-surface are