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f, ϕ : Ω → R ¯x
f ϕ = 0 ϕ(¯x) = 0 ¯x f |ϕ=0
f ϕ = 0 f
ϕ = 0
Ω ⊆ R n Γ ⊆ Ω f : Ω → R
g : V → Γ V ⊆ R m m ≤ n C 1
f Γ g f ◦ g
V V int V V ∩ ∂V
Ω ⊆ R n f :
R n → R C 1 (Ω) ϕ : Ω → R C 1 ¯x ∈ Ω f
ϕ = 0 Dϕ(¯x) = 0 λ ∈ R
Df(¯x) + λDϕ(¯x) = 0.
¯x ∈ Ω f ϕ = 0 ¯x
Γ := {x ∈ R n : ϕ(x) = 0}, Cx0 := {x ∈ Rn : f(x) = f(x0)}.
f(x, y) = x 3 + 4xy 2 − 4x
x 2 + y 2 − 1 = 0
g(x, y) = x 2 + y 2 − 1 ∇g(x, y) = (0, 0)
x = y = 0 g(x, y) = 0 g(0, 0) = 0
L(x, y, λ) := f(x, y) + λg(x, y) = x 3 + 4xy 2 − 4x + λ(x 2 + y 2 − 1).
∂xL(x, y, λ) = 3x 2 + 4y 2 − 4 + 2λx
∂yL(x, y, λ) = 8xy + 2λy = 2y(λ + 4x).
∂xL(x, y) = ∂yL(x, y) = 0 y = 0 4x = −λ
y = 0 x = ±1
y = 0 x = −λ/4 y 2 = 1 − λ 2 /16
0 ≤ λ ≤ 4
3 λ2
+ 4
16
1 − λ2
16
− 4 + 2λ
− λ
= 0,
4
λ = 0 x = 0 y = ±1 (±1, 0) (0, ±1)
f(1, 0) = −3 f(−1, 0) = 3 f(0, ±1) = 0 (−1, 0)
(1, 0)