162
Appendix A: Geometric Quantities
Fig. A.3 Curvilinear
coordinates for a spherical
structure
β
X
3
α
X
1
X
2
Θ
2
Θ
1
Θ
3
The position vectors of an arbitrary point in the shell space and at the mid-surface
are respectively expressed as
R = (R + Θ
3
)
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sin
Θ
1
R
cos
Θ
2
sin
Θ
1
R
cos
Θ
2
cos
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, r = R
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sin
Θ
1
R
cos
Θ
2
sin
Θ
1
R
cos
Θ
2
cos
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
.
(A.28)
The covariant base vectors for an arbitrary point in the shell space are
g 1 =
1 +
Θ
3
R
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
cos
Θ
1
R
cos
Θ
2
cos
Θ
1
R
sin
Θ
2
− sin
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
g 2 = (R + Θ
3
)
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
− sin
Θ
1
R
sin
Θ
2
sin
Θ
1
R
cos
Θ
2
0
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
, g 3 =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sin
Θ
1
R
cos
Θ
2
sin
Θ
1
R
sin
Θ
2
cos
Θ
1
R
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
.
(A.29)
The covariant metric tensor in the shell space is
g i j = g i · g j =
⎡
⎢
⎢
⎢
⎢
⎣
1 +
Θ
3
R
2
0
0
0
R + Θ
3
2 sin
2
Θ
1
R
0
0
0
1
⎤
⎥
⎥
⎥
⎥
⎦
.
(A.30)