S 11 ¼
1
A 1 1 þ
α 3
R 1
∂u 1
∂α 1
þ
u 2
A 2
∂A 1
∂α 2
þ
A 1 u 3
R 1
!
,
S 22 ¼
1
A 2 1 þ
α 3
R 2
∂u 2
∂α 2
þ
u 1
A 1
∂A 2
∂α 1
þ
A 2 u 3
R 2
!
, S 33 ¼
∂u 3
∂α 3
,
ð5:109Þ
2S 23 ¼ A 2 1 þ
α 3
R 2
∂
∂α 3
u 2
A 2 1 þ
α 3
R 2
2
4
3
5 þ
1
A 2 1 þ
α 3
R 2
∂u 3
∂α 2
ð5:110Þ
2S 31 ¼ A 1 1 þ
α 3
R 1
∂
∂α 3
u 1
A 1 1 þ
α 3
R 1
2
4
3
5 þ
1
A 1 1 þ
α 3
R 1
∂u 3
∂α 1
ð5:111Þ
2S 12 ¼
A 1 1 þ
α 3
R 1
A 2 1 þ
α 3
R 2
∂
∂α 2
u 1
A 1 1 þ
α 3
R 1
2
4
3
5 þ
A 2 1 þ
α 3
R 2
A 1 1 þ
α 3
R 1
∂
∂α 1
u 2
A 2 1 þ
α 3
R 2
2
4
3
5 :
ð5:112Þ
The gradient of a scalar field, e.g., the electric potential, in the shell coordinates is
given by
Fig. 5.3 A differential
element of a thin shell and
coordinate system
5.11 Equations for Shells
133