166
Appendix A: Geometric Quantities
Table A.1 Notations of frequently used geometric quantities
Notation
Sphere
Cylinder
Plate
a 1 = a 11
1
1
1
a 2 = a 22
1
R 2 sin 2
Θ 1
R
1
R 2
1
b 1 = b 11
−
1
R
0
0
b 2 = b 22
−R sin 2
Θ 1
R
−R
0
t 1 = −Γ 2
12 = −Γ 2
21
−
cos
Θ 1
R
R sin
Θ 1
R
0
0
t 2 = −Γ 1
22
R sin
Θ 1
R
cos
Θ 1
R
0
0
c 1 = b 1
·1
c 1 = a 1 b 1
c 2 = b 2
·2
c 2 = a 2 b 2
Using the notations introduced in Table A.1, a general expression of
n
ϕ αβ for the
three curvilinear coordinate systems can be obtained as
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n
ϕ 11 =
n
v 1,1 − b 11
n
v 3
=
n
v 1,1 − b 1
n
v 3
n
ϕ 12 =
n
v 1,2 − Γ
2
12
n
v 2
=
n
v 1,2 + t 1
n
v 2
n
ϕ 21 =
n
v 2,1 − Γ
2
21
n
v 2
=
n
v 2,1 + t 1
n
v 2
n
ϕ 22 =
n
v 2,2 − Γ
1
22
n
v 1 − b 22
n
v 3
=
n
v 2,2 + t 2
n
v 1 − b 2
n
v 3
n
ϕ 31 =
n
v 3,1 + b
1
·1
n
v 1
=
n
v 3,1 + c 1
n
v 1
n
ϕ 32 =
n
v 3,2 + b
2
·2
n
v 2
=
n
v 3,2 + c 2
n
v 2
(A.41)