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I g(m1, m2, m3) = 0 g(m1, m2, m3) = m3 − m 1/3 1 m1/2 2 m1, m2, m3 ≥ 0 p1 p2 h(m1, m2) := I(m1, m2, m 1/3 1 m1/2 2 , p1, p2) = m 1/3 1 m1/2 2 − p1m1 − m2p2, m1, m2 ≥ 0. mi i = 1, 2 m 1/3 1 ≤ p1m1/2 m 1/2 2 ≤ p2m2/2 |(m1, m2)| h(m1, m2) ≤ −p1m1/2−m2p2/2 |(m1, m2)| → +∞ h(m1, m2) → −∞ R > 0 |(m1, m2)| > R h(m1, m2) ≤ h(0, 0) = 0 (m1, m2) /∈ B(0, R) h (m1, m2) h (1, 1) ∈ B(0, R) B(0, R) ∩ {m1 ≥ 0, m2 ≥ 0} h h(0, m2) = −p2m2 h(m1, 0) = −p1m1 h(0, 0) = 0 ( ¯m1, ¯m2) ( ˜m1, ˜m2) {m1, m2 > 0} (1−λ)( ¯m1, ¯m2)+λ( ˜m1, ˜m2) λ[−ε, 1+ε] ε > 0 {m1, m2 > 0} ψ(λ) := h((1 − λ)( ¯m1, ¯m2) + λ( ˜m1, ˜m2)), d dλ ψ(λ) = ∇h((1 − λ)( ¯m1, ¯m2) + λ( ˜m1, ˜m2)) · ( ˜m1 − ¯m1, ˜m2 − ¯m2), d 2 dλ 2 ψ(λ) = 〈D2 h((1 − λ)( ¯m1, ¯m2) + λ( ˜m1, ˜m2))( ˜m1 − ¯m1, ˜m2 − ¯m2), ( ˜m1 − ¯m1, ˜m2 − ¯m2)〉 d d dλψ(0) = dλψ(1) = 0 ( ¯m1, ¯m2) ( ˜m1, ˜m2) h(m1, m2) D 2 ⎛ − ⎜ h(m1, m2) = ⎜ ⎝ 2 9 m−5/3 1 m1/2 1 2 6 m−2/3 1 m−1/2 2 1 6 m−2/3 1 m−1/2 2 − 1 4 m1/3 1 m−3/2 ⎞ ⎟ ⎠ 2 . . m −4/3 1 /(36m2) > 0 d2 dλ2 ψ(λ) < 0 ψ ( ¯m1, ¯m2) = ( ˜m1, ˜m2) ∇h(m1, m2) = (0, 0) ⎧ 1 ⎪⎨ 3 m−2/3 1 m1/2 2 − p1 = 0, ⎪⎩ 1 2 m1/3 1 m−1/2 2 − p2 = 0, p1 = 1 3 m−2/3 1 m1/2 2 p2 = 1 2 m1/3 1 m−1/2 2 I m1, m2, m 1/3 1 m1/2 2 , 1 3 m−2/3 1 m1/2 2 , 1 2 m1/3 1 m−1/2 2 = 1 6 m1/3 1 m1/2 2 > 0. {m1, m2 ≥ 0}
⎛ 1 ⎜ 3 F (m1, m2, p1, p2) = ⎜ ⎝ m−2/3 1 m1/2 2 − p1, 1 2 m1/3 1 m−1/2 ⎞ ⎟ ⎠ 2 − p2, . F (m1, m2, p1, p2) = 0. Dm1,m2 F F (m1, m2) h m1 = m1(p1, p2) m2 = m2(p1, p2) m1 m2 p1, p2 s(p1, p2) = ((p1, p2), m2(p1, p2)) Jac s(p1, p2) = −[Dm1,m2 F (m1, m2, p1, p2)] −1 |(m1,m2)=s(p1,p2) ◦ D[p1, p2]F (m1, m2, p1, p2) |(m1,m2)=s(p1,p2) ⎛ − ⎜ = − ⎜ ⎝ 2 9 m−5/3 1 1 6 m−2/3 1 ⎛ − ⎜ = ⎜ ⎝ 2 9 m−5/3 1 1 6 m−2/3 1 2 ⎛ = ⎜ ⎝ m1/2 2 m−1/2 2 m1/2 2 m−1/2 −9 m 5/3 1 m−1/2 2 −6 m 2/3 1 m1/2 2 1 6 m−2/3 1 m−1/2 2 − 1 4 m1/3 1 m−3/2 2 1 6 m−2/3 1 m−1/2 2 − 1 4 m1/3 1 m−3/2 2 −6 m 2/3 1 m1/2 2 −8 m −1/3 1 m3/2 2 ∂ m1(p1, p2) = −9 m ∂p1 5/3 ∂ ∂p2 ∂ ∂p1 ∂ ∂p2 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ −1 |(m1,m2)=s(p1,p2) −1 |(m1,m2)=s(p1,p2) |(m1,m2)=s(p1,p2) 1 (p1, p2)m −1/2 2 (p1, p2) < 0, m1(p1, p2) = −6 m 2/3 1 (p1, p2)m 1/2 2 (p1, p2) < 0, m2(p1, p2) = −6 m 2/3 1 (p1, p2)m 1/2 2 (p1, p2) < 0, m2(p1, p2) = −8 m −1/3 1 (p1, p2)m 3/2 2 (p1, p2) < 0. −1 0 · 0 −1 mi = mi(p1, p2) i = 1, 2
Page 3 and 4:
R R n
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2 × 2 SO(
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R a
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R F G R F G ¯ F ⊇ G
Page 14 and 15:
R n x, y ∈ Rn x = (x1, ...
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(X, τ) D ⊆ X X
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f : D → R D ⊆ Rn γ : [
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A =]0, +∞[×]0, +∞[ f : A →
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lim (x,y)→(0,0) (x,y)=(0,0) |xy|
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E E E f −1 ([1, +∞[)
Page 28 and 29:
s ↦→ 1 − e −s s ≥ 0 1
Page 30 and 31:
Fn : [0, +∞[→ R Fn(ρ) = 2n ρ
Page 33 and 34: I =]a, b[ R {fn}n∈N fn : I
Page 35: K [0, +∞[ R > 0 B(0, R) ⊇
Page 38 and 39: an = 0 n = 2k an = 4/(πn2 ) n
Page 40 and 41: 5 1 18 = + 2 4 π2 ∞ n=0 ∞ n=0
Page 43 and 44: g(x) := x(π−x) x ∈ [0, π]
Page 45 and 46: X Y T : X → Y T ℓ >
Page 47 and 48: X K E X a, b ∈ R f : E × [a
Page 49 and 50: |f(x, y)| ≤ log(1 + 3y
Page 51 and 52: (x, y) ∈ R 2 xy =
Page 53 and 54: X, Y D ⊆ X f : D → Y u ∈
Page 55 and 56: H A 2 × 2 λ 2
Page 57: (2kπ, π/2+2hπ) (π+2kπ, 3π/2+2
Page 60 and 61: ∂xf(x, y, z) = −2 cos x sin x =
Page 62 and 63: (0, 0) f : R 2 → R f(x, y) =
Page 65 and 66: α ∈ R (0, 0) f(x, y) = 2
Page 67: Cα > 0 α /∈ [−1/2, −1/4]
Page 70 and 71: x = cos θ y = sin θ θ ∈ [−π
Page 72 and 73: γ(t) = (t, 1/t) t → ±∞ L(x
Page 74 and 75: ⎧ ⎪⎨ 1 + λ(2x − 2) = 0, 1
Page 76 and 77: f(x, y) = x 2 + y 2 V = {(x, y)
Page 78 and 79: ⎧⎪ 2x + λ(2x + y) + 2µx = 0
Page 81 and 82: R 2 D ⊆ R 2 f : D → R (x0,
Page 83: x = 1 y = −1 z = 2 y z
Page 88 and 89: k = 3√ 4 ξ ζ
Page 90 and 91: x = ρ cos θ y = ρ sin θ ρ 2 (
Page 92 and 93: f(x, y, z) = ez+x2 f + αx + y
Page 94 and 95: ϕ(1) = 0 ϕ ′ (1) = − ∂xf(1
Page 96 and 97: y 3 − xy 2 + x 2 y = x − x 3
Page 98 and 99: X = {(ρ, θ) ∈ R 2 : ρ > 0, 0 <
Page 100 and 101: arcsin sin θ = θ θ ∈] − π
Page 102 and 103: I R r : I → Rd γ µ : r
Page 104 and 105: γ lunghezza(γ) = |r ′ (θ)|
Page 106 and 107: p = 2 |(x, y)| −p dx dy = R 2 \B
Page 108 and 109: ρ 3 cos θ = aρ 2 (cos 2 θ − s
Page 111 and 112: Γ := {(x, y) ∈ R 2 : (x 2 + y 2
Page 113 and 114: x Γ Q2 Q3 x Γ \
Page 115 and 116: (x 2 + y 2 )(y 2 + x(x + 1)) = 4xy
Page 117 and 118: 4 ±1, ±2, ±4 ±1, ±2 4
Page 119 and 120: −2 + 8t − 4t 2 t = 1/2(2
Page 121 and 122: a ∈ R R 2 (x 2 + y 2 ) 3 = 4
Page 123: (x 2 + y 2 ) 3 = 4x 2 y 2
Page 127 and 128: a > 0 x = at at2 , y = . 1
Page 129: cos(π − α) = − cos(α) π/4
Page 132 and 133: I := F · ˆn dσ ˆn Σ =
Page 134 and 135:
S ψ(x, y) = (ψ1, ψ2, ψ3) = (
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C = {(x, y, z) : (x, y) ∈ D, λ <
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ˆn = ∇G = ((∇G)1, (∇G)2, (
Page 140 and 141:
2π 0 cos θ sin θ dθ = 1 2 2π
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L − F · ˆn dσ = − L− √
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K ⊆ R n χK : R n → {0, 1}
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f [1, +∞[ [0, 1] [0
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f α k = 1 x α + k α , sα n = n
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D R X n 1 1 X C ℓ
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γ2(1/2) = (1/8, 3/4) γ2(1) = (0,
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C = −B ∂yu(x, y) = C/x 1 + y2 y
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dy N(x, y) = − dx M(x, y) N M
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2xy dx + (x 2 + 1) dy = 0 (x 2 + y
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x − c = arcsin(2y − 1) 0 < y
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h(x, y) = 1/(|x| √ x2 − 1)
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˙x = e t−x /x x(α) = 1 α ∈ R
Page 168 and 169:
˙x = x 2 /(1 − tx) x(0) = α α
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y ′′′ − 6y ′′ + 11y ′
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c ′ 1 (x) cos x + c′ 2 (x) sin
Page 175 and 176:
y ′ + y 1 x = tan x sin x y ′
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λ 2 − 4λ − 5 = 0, A
Page 179 and 180:
F (x, y, ˙y) = 0, F : A → R A
Page 181 and 182:
F (x, y, p) = 0 c = 0 dy = p dx
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y = xy ′ + f(y ′ ) F (x, y, p
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y → 0 −∞
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x > 0 x y = ± 1 + x x < 0
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[0, π] utt + 2ut − uxx = 0,
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e |uk(x, t)| = C −t sin( 4k(1
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∞ n=1 bne −x sin nx = x(
Page 198 and 199:
T ˙ Tn(t) = −5n 2 Tn(t) Tn(
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⎧ −utt + 3uxx = 0 ]0, π[×]0,
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∆ = 0 λ = −1 µ1 = µ2 = −
Page 204 and 205:
x = cos θ y = sin θ F F (co
Page 206 and 207:
I := arctan y dx − xy dy γ γ
Page 208 and 209:
f(ρ) f ′ (ρ) = 4ρ3 + 36ρ 8
Page 210 and 211:
Jac ϕ(θ, y) = ⎛ ⎜ ⎝ − y
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z = (x, y) ˙z = Az + B(t)
Page 214 and 215:
Q0 Q2 Q1 Q1
Page 216 and 217:
˙x = t − x 2 x(0) = a a ∈ R
Page 219 and 220:
2π xt sin t F (x) = e dt x ∈ R
Page 221 and 222:
xz 3 dx dy dz, T T y
Page 223 and 224:
α > 0 lim (x,y)→(0,0) (x,y)=(0,0
Page 225 and 226:
F (0, 1) = 0 ∂yF (x, y) = 3y 2
Page 227 and 228:
(u + v) D 2 + (u − v) 2 1 + 2(u2
Page 229 and 230:
α = = 2 3 π. 2π 0
Page 231 and 232:
F D z = 0 (0,
Page 233 and 234:
y ′ = 2xy − x y(0) = 3/2
Page 235 and 236:
Γ f(x, y) = 0 f : R 2 → R
Page 237 and 238:
∇f(x0, y0) = (0, 0) (x0, y0)
Page 239 and 240:
ϕ : I ×J → R 3 I, J R
Page 241 and 242:
ψ(a, t) = ψ(b, t) t ∈]c, d[
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[
Page 245 and 246:
Y K Ω R × Ω I R f :
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1 ω(x, y) = M(x, y) dx + N(x, y)
Page 249 and 250:
M(x, y) dx + N(x, y) dy = 0 1 M, N
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x, y z dy dz
Page 254 and 255:
f(x) = N(x) Ak1 = + D(x) x − xk
Page 256 and 257:
Φ y ′ = A(t) y ′ = A(t)
Page 258 and 259:
det P = 4 √ 7 = 0 P −1 AP
Page 260 and 261:
y ′′ (t) + py ′ (t) + qy(t) =
Page 262 and 263:
y(0) = c1 + d1 = 1 ˙y(t) = (−c1
Page 264 and 265:
a ′ x + b ′ y + c ′ = 0
Page 266 and 267:
η2 g g(η) > 0 y0 < η <
Page 269 and 270:
2 × 2 F ∈ R[x] n
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2 × 2 t ↦→ x(
Page 274 and 275:
∆ < 0 α = −d 2c ω = √ |
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a = 0 b = 0 c = 0 ⎧ ⎪⎨ b
Page 279 and 280:
SO(3) U(3) := {O ∈
Page 281:
SO(3) Aξ = ω × ξ ξ
Page 284 and 285:
x 2 + y 2 = 1 x 2 − y 2 = 1
Page 286 and 287:
1 arcsin(x) + x − x2 dx = √ 1
Page 288 and 289:
z 1 + cos z cos = ± 2 2 z 1
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