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ω2 = det 2 B1 + det 2 B2 + det 2 B3
cos θ ρ sin θ
B1 =
, det
sin θ ρ cos θ
2 B1 = ρ 2 .
cos θ ρ sin θ
B2 =
2ρ − 4ρ3
, det
0
2 B2 = ρ 2 (2ρ − 4ρ 3 ) 2 sin 2 θ,
sin θ ρ cos θ
B3 =
2ρ − 4ρ3
, det
0
2 B3 = ρ 2 (2ρ − 4ρ 3 ) 2 cos 2 θ.
ω2 = ρ2 + ρ2 (2ρ − 4ρ3 ) 2 = ρ 1 + 4ρ2 + 16ρ6 − 16ρ4 .
Σ = {(x, y, 0) : x2 +y2 ≤ α2 } ˆn = (0, 0, −1)
Σ ∪ S Ω S Σ Ω
div F dxdydz = Φ(S, F ) + Φ(Σ, F ).
α
0
Ω
Σ F (x, y, 0) · (0, 0, −1) = 0
Φ(Σ,
F ) = F · ˆn dσ = 0,
div
Ω
F dxdydz = dxdydx = Volume(Ω).
Ω
Ω
Σ
ψ(ρ, θ, z) = (ρ cos θ, ρ sin θ, z),
0 < ρ < α 0 < z < ρ 2 (1 − ρ 2 )
ρ Ω
2π α ρ2 (1−ρ2 )+1
α
ρ dzdρdθ = 2π (ρ
0 0 0
0
2 (1 − ρ 2
α2 ) + 1)ρ dρ = 2π
2
⎛
2π
det ⎝
0
F1 ◦ ϕ cos θ ρ sin θ
F2 ◦ ϕ sin θ ρ cos θ
F3 ◦ ϕ 2ρ − 4ρ3 ⎞
⎠ dθ dρ =
0
⎛
⎞
=
=
α 2π
e
det ⎝
0 0
4ρ sin θ ρ cos θ − ρ
cos θ ρ sin θ
2 cos2 θ sin θ ρ cos θ
ρ2 (1 − ρ2 ) + 1 2ρ − 4ρ3 ⎠ dθ dρ
α 2π
(ρ
0
0 0
2 (1 − ρ 2 ) + 1)ρ dρdθ+
α 2π
− (e
0 0
4ρ sin θ ρ cos θ − ρ cos θ − ρ 2 cos 2 θρ sin θ) dθ dρ
α2 α 2π
α4 α6
1 d
+ − −
2 4 6 0 0 4 dθ (e4ρ sin θ α 2π
) dθdρ +
0 0
= 2π
α2 α4 α6
= 2π + − ,
2 4 6
+ α4
4
ρ 3
3
α6
− .
6
d
dθ (cos3 θ) dθdρ