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S − θ ∈ [−π/4, π/4]
(0, 0, −1) ( √ 2, 0, −1) y = 0 z
C (0, 0, 1)
S − = {(r cos θ, r sin θ, 1) ∈ R 3 : 0 ≤ r ≤ √ 2, |θ| ≤ π/4}
L + L −
L + = {(r cos θ, r sin θ, z) : θ = π/4, 0 ≤ r ≤ 1 + 2z 2 , |z| ≤ 1},
L − = {(r cos θ, r sin θ, z) : θ = π/4, 0 ≤ r ≤ 1 + 2z2 , |z| ≤ 1}.
(− √ 2/2, √ 2/2, 0) (− √ 2/2, − √ 2/2, 0)
Σ C (1, 0, 0) (1, 0, 0)
Σ
− F · ˆn dσ =
F · ˆn dσ,
Σ
S + ∪S − ∪L + ∪L −
S + ∪S − ∪L + ∪L −
F · ˆn dσ,
C (x, y, z) ∈ L + F · ˆn = 0
L + F (x, y, 1) = F (x, y, −1) (x, y, 1) ∈ S +
(x, y, −1) ∈ S− ˆn C F ·ˆn(x, y, 1) = − F ·ˆn(x, y, −1)
(x, y, 1) ∈ S + (x, y, −1) ∈ S−
S +
F · ˆn dσ +
F · ˆn dσ = F3(x, y, z) dσ =
S +
S +
S −
F · ˆn dσ = 0.
S +
(x 2 + y 2
) dσ =
√ 2 π/2
0 −π/2
F · ˆn dσ.
(x, y, z) ∈ L −
L −
ψ(r, z) =
L −
√
F
2
· ˆn = −√
1 + 2z2 .
√
2
2 r,
√
2
r, z , z ∈ [−1, 1], r ∈ [0,
2 1 + 2z2 ].
⎛
Jac ψ(r, z) = ⎝
√ 2/2 0
√ 2/2 0
0 1
⎞
⎠
r 2 r drdθ = π
2 .
2
2
ω2(∂rψ, ∂zψ) = det 2
√
√
2/2 0
+ det
2/2 0
2
√
2/2 0
+ det
0 1
2
√
2/2 0
= 1.
0 1