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Γ := {(x, y) ∈ R 2 : (x 2 + y 2 )(y 2 + x(x + 1)) = 4xy 2 }
cos(3θ) = cos θ(1 − 4 sin2 θ)
Γ Γ
2π
3
Γ P1 = (−1, 0) P2 = (1/2, √ 3/2)
P3 = (1/2, − √ 3/2)
Γ P0 y = ϕ1(x) ϕ1(−1) = 0
ϕ ′ 1 (0)
Γ P1 y = ϕ2(x) ϕ2(1/2) =
√ 3/2 ϕ ′ 2 (1/2)
Γ P2 y = ϕ3(x) ϕ3(1/2) =
− √ 3/2 ϕ ′ 3 (1/2)
h(x, y) = x 2 + y 2 Γ Γ
Γ
x y ↦→ −y
f(x, y) = (x 2 + y 2 )(y 2 + x(x + 1)) − 4xy 2 .
cos 3θ = cos(θ + 2θ) = cos θ cos 2θ − 2 sin 2 θ cos θ
= cos θ(1 − 2 sin 2 θ) − 2 sin 2 θ cos θ = cos θ(1 − 4 sin 2 θ)
Γ x = ρ cos θ y = ρ sin θ
f(ρ cos θ, ρ sin θ) = ρ 2 (ρ 2 + ρ cos θ) − 4ρ 3 cos θ sin 2 θ = ρ 3 (ρ + cos θ(1 − 4 sin 2 θ))
ρ = 0 ρ + cos θ(1 − 4 sin 2 θ) = 0
θ = π/2
Γ = {(ρ cos θ, ρ sin θ) : ρ = − cos 3θ, ρ ≥ 0, θ ∈ [0, 2π]}.
cos 3θ = cos(3(θ + 2π/3))
f
df(x, y) = (2x(y 2 + x(x + 1)) + (x 2 + y 2 )(2x + 1) − 4y 2 ) dx+
+ (2y(y 2 + x(x + 1) + x 2 + y 2 ) − 8xy) dy
= (4x 3 + 3x 2 + 4xy 2 − 3y 2 ) dx + 2y(2y 2 + 2x 2 − 3x) dy