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x = ρ cos θ y = ρ sin θ ρ 2 (ρ 2 cos 4 θ − 4 cos θ sin θ + ρ 2 sin 4 θ) = 0
ρ = 0
ρ 2 =
4 cos θ sin θ
cos 4 θ + sin 4 θ .
cos θ sin θ > 0 Γ
ρ(θ) =
4 cos θ sin θ
cos 4 θ + sin 4 θ
cos 4 θ + sin 4 θ x 4 + y 4
x 2 + y 2 = 1 x 4 + y 4
ρ
Γ
ρ 0 (0, 0) ∈ Γ lim θ→0 + ρ(θ) = 0
lim
θ→0
θ→π/2
θ→π
θ→3/2π
ρ(θ) = lim ρ(θ) = lim ρ(θ) = lim ρ(θ) = 0.
+ − + −
y 2 Γ ρ 2 (θ) sin 2 θ θ =
0, π/2, π, 3/2π y = 0 0 < θ < π/2
f1(θ) = ρ 2 (θ) sin 2 θ = 4 cos θ sin3 θ
cos 4 θ + sin 4 θ = 4 tan3 θ
1 + tan 4 θ .
g(s) = 4s 3 /(1 + s 4 ) s > 0
g ′ (s) = 4 3s2 (1 + s 4 ) − 4s 6
(1 + s 4 ) 2
= 4s2 (3 − s4 )
(1 + s4 .
) 2
s = 4√ 3
tan :]0, π/2[→]0, +∞[
f1 = g ◦ tan θm = arctan 4√ 3 f1 [0, π/2]