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f(ρ) <br />
f ′ (ρ) = 4ρ3 + 36ρ<br />
8ρ 3<br />
− 3 ρ 4 + 18ρ 2 − 27 <br />
8ρ 4<br />
= (−9 + ρ2 ) 2<br />
8r 4<br />
ρ = 3 f ρ <br />
cos 3θ = −1 θ1 = π/3 θ2 = 5π/3 θ3 = π cos 3θ = 1 <br />
θ = 0 θ = 2/3π θ = 4π/3 ρ f(ρ) = −1 <br />
(ρmin, π) <br />
ρ f(ρ) = 1 <br />
(ρmax, 0) f(ρ) = −1 <br />
ρ 4 + 8ρ 3 + 18ρ 2 − 27 = 0<br />
27 ±1, ±3, ±9 <br />
1 <br />
ρ 4 + 8ρ 3 + 18ρ 2 − 27 = (ρ − 1)(ρ 3 + bρ 2 + cρ + d) = ρ 4 + (b − 1)ρ 3 + (c − b)ρ 2 + (d − c)ρ − d)<br />
b = 9 c = 27 d = 27 <br />
≥ 0,<br />
ρ 4 + 8ρ 3 + 18ρ 2 − 27 = (ρ − 1)(ρ 3 + 9ρ 2 + 27ρ + 27) = (ρ − 1)(ρ + 3) 3<br />
ρ = 1 ρ <br />
f(ρ) = +1 <br />
ρ 4 − 8ρ 3 + 18ρ 2 − 27 = 0<br />
27 ±1, ±3, ±9 <br />
−1 <br />
ρ 4 − 8ρ 3 + 18ρ 2 − 27 = (ρ + 1)(ρ 3 + bρ 2 + cρ + d) = ρ 4 + (b + 1)ρ 3 + (c + b)ρ 2 + (d + c)ρ + d)<br />
b = −9 c = 27 d = −27 <br />
ρ 4 + 8ρ 3 + 18ρ 2 − 27 = (ρ + 1)(ρ 3 − 9ρ 2 + 27ρ − 27) = (ρ − 1)(ρ − 3) 3 ,<br />
ρ = 3 ρ <br />
ρ 2 9<br />
(3, 0) ∂xF (3, 0) =<br />
∂yF (3, 0) = 0 (3, 0) 2π/3 <br />
0 <br />
<br />
R 3 S <br />
ϕ(θ, y) = ( y 2 + 1 cos θ, y, y 2 + 1 sin θ), θ ∈ [0, 2π], |y| < 1,<br />
F : R 3 → R 3 F (x, y, z) = (x 2 , y/2, x)<br />
F F <br />
F <br />
√ 2 (0, 1, 0) y = 1 <br />
γ(θ) = ( √ 2 cos θ, 1, √ 2 sin θ), θ ∈ [0, 2π].<br />
ϕ 2 <br />
ϕ<br />
(1, 0, 0)<br />
F S <br />
<br />
ϕ(θ, y) = (ϕ1, ϕ2, ϕ3) F = (F1, F2, F3)
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