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g1(x, y, z) = x 2 +y 2 −1 g2(x, y, z) = x+y+z−1 =
0 g −1
1 (0)∩g−1 2 (0)
∇g1(x, y, z) = 0 x = y = 0 g1(x, y, z) = 0 ∇g2(x, y, z) = 0
x 2 +y 2 = 1 |x| ≤ 1 |y| ≤ 1
x + y + z = 1 z = 1 − x − y |z| ≤ 3
f
L(x, y, z, λ, µ) = f(x, y, z) + λg1(x, y, z) + µg2(x, y, z) ∇L = 0
⎧
∂xf(x, y, z) + λ∂xg1(x, y, z) + µ∂xg2(x, y, z) = 0
⎪⎨ ∂yf(x, y, z) + λ∂yg1(x, y, z) + µ∂yg2(x, y, z) = 0
∂zf(x, y, z) + λ∂zg1(x, y, z) + µ∂zg2(x, y, z) = 0
g1(x, y, z) = 0
⎪⎩
g2(x, y, z) = 0
⎧
−1 + 2x + y + 2λx + µ = 0
⎪⎨ −1 + x + 2y + z + 2λy + µ = 0
y + µ = 0
x 2 + y 2 = 1
⎪⎩
x + y + z = 1
y = −µ −1+2x+2λx = 0 2x(1+λ) = 1
λ = −1 x = 1
2(1+λ)
λy = 0 y = 0 λ = 0 λ = 0 x = 1/2 y = ± √ 3/2
y = 0 x = ±1 (1, 0, 0)
(−1, 0, 2) (1/2, √ 3/2, 1/2 − √ 3/2) (1/2, − √ 3/2, 1/2 + √ 3/2) f(1, 0, 0) = 0 f(−1, 0, 2) = 2
f(1/2, √ 3/2, 1/2 − √ 3/2) = f(1/2, − √ 3/2, 1/2 + √ 3/2) = −1/4 (−1, 0, 2)
(1/2, √ 3/2, 1/2 − √ 3/2) (1/2, − √ 3/2, 1/2 + √ 3/2)
R 3
V V = {(x, y, z) : x 2 + y 2 + xy − z 2 = 1 ∧ x 2 + y 2 = 1}
g1(x, y, z) = x2 + y2 + xy − z2 − 1 g2(x, y, z) =
x2 + y2 − 1 = 0
g −1
1
(0) ∩ g−1
2 (0) ∇g1(x, y) = (2x + y, 2y + x, −2z) x = y = z = 0
g1(x, y, z) = 0 ∇g2(x, y) = 0 x = y = 0 g2(x, y) = 0
x 2 +y 2 = 1 |x| ≤ 1
|y| ≤ 1 x 2 +y 2 +xy−z 2 = 1 z 2 = 1+x 2 +y 2 +xy |z| ≤ 2
f(x, y, z) = x 2 +y 2 +z 2
L(x, y, z, λ, µ) = f(x, y, z) + λg1(x, y, z) + µg2(x, y, z)
∇L = 0
⎧
∂xf(x, y, z) + λ∂xg1(x, y, z) + µ∂xg2(x, y, z) = 0
⎪⎨ ∂yf(x, y, z) + λ∂yg1(x, y, z) + µ∂yg2(x, y, z) = 0
∂zf(x, y, z) + λ∂zg1(x, y, z) + µ∂zg2(x, y, z) = 0
g1(x, y, z) = 0
⎪⎩
g2(x, y, z) = 0