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(u + v)
D
2 + (u − v) 2
1 + 2(u2 + v2 ) 2 1 + u2 + v2 dudv
= 4 (u 2 + v 2 ) 1 + 2(u2 + v2 ) 1 + (u2 + v2 ) dudv
= 4
(w = ρ 2 ) = 4π
= 4π
D
2π 1
0
1
0
1
0
0
ρ 2 1 + 2ρ 2 1 + ρ 2 ρ dρdθ
w (1 + 2w)(1 + w) dw
w 2w 2 + 3w + 1 dw
= 4π(44 + 132 √ 6 − 9 √ 2 log(3 + 2 √ 2) + 9 √ 2 log(7 + 4 √ 3)),
√ s 2 + 1 1/2(y 1 + y 2 + log(z + √ 1 + z 2 ))
a ≥ 0 R 3 Sa
ϕ(θ, y) = ( y 2 + a 2 cos θ, y, y 2 + a 2 sin θ), θ ∈ [0, 2π], |y| < 1,
F : R 3 → R 3 F (x, y, z) = (x 2 , y/2, x)
F
F
√ 1 + a 2 y = 1
γ(θ) = ( a 2 + 1 cos θ, 1, a 2 + 1 sin θ), θ ∈ [0, 2π].
F
ϕ 2
ϕ Sa
a = 0
a > 0 (a, 0, 0)
F Sa
ϕ(θ, y) = (ϕ1, ϕ2, ϕ3) F = (F1, F2, F3)
div F (x, y, z) = ∂xF1 + ∂yF2 + ∂zF3 = 2x + 1/2,
⎛
rot F = det ⎝ e1 ∂x x2 e2 ∂y y/2 ⎠ = (0, −1, 0).
e3 ∂z x
D =
{(x, 1, z) : x2 +z 2 ≤ (1+a 2 )} (0, −1, 0) (0, −1, 0) D
γ
rot
F · ˆn dσ = dσ = Area(D) = π(1 + a 2 ).
D
D
2π
F dγ = F ( a2 + 1 cos θ, 1, a2 + 1 sin θ) · (− a2 + 1 sin θ, 0, a2 + 1 cos θ) dθ
γ
=
0
2π
0
− (a 2 + 1) 3/2 cos 2 θ sin θ + (a 2 + 1) cos 2
θ dθ = π(1 + a 2 )
F
⎞