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df(x, y) = ∂xf(x, y) dx + ∂yf(x, y) dy = 2(x − y) dx + 2(3y 2 − x) dy. ∇f(x, y) = 2(x − y, 3y 2 − x) S = {(x, y) ∈ R 2 : f(x, y) = 0} (x, y) ∈ S C ∈ R C + df(x, y)(h, k) = 0 (h, k) (x, y) (2, 1) C + 2h − k = 0 (2, 1) h = 2 k = 1 C = −3 (2, 1) k = 2h − 3 ϕ ′ (2) = 2 xe y − y + 1 = 0, x = 0 y = ϕ(x) x = 0 1 f(x, y) = xe y − y + 1 f ∈ C ∞ (R 2 ) f(0, 1) = 0 ∂xf(x, y) = e y ∂yf(x, y) = xe y − 1 ∂yf(0, 1) = −1 = 0 ϕ f(x, y) = 0 (0, 1) ϕ(0) = 1 C 1 f ∈ C 1 ϕ ′ (0) = e ϕ ′ (x) = − ∂xf(x, ϕ(x)) xe y − 1 = − eϕ(x) xe ϕ(x) − 1 . ϕ ′′ (x) = − eϕ(x) ϕ ′ (x)(xe ϕ(x) − 1) − e ϕ(x) (e ϕ(x) + xe ϕ(x) ϕ ′ (x)) (xe ϕ(x) − 1) 2 ϕ ′′ (0) = e 2 x 2 y − z 3 + 4xy 3 z + x 3 z + 30 = 0 = e2ϕ(x) (1 − xϕ ′ (x)) (xeϕ(x) − 1) 2 , (1, 2) z = ϕ(x, y) x = 1 y = 2 −1 ϕ f(x, y, z) = x 2 y − z 3 + 4xy 3 z + x 3 z + 30 f(1, 2, −1) = 2 + 1 − 32 − 1 + 30 = 0 f ∂xf(x, y, z) = 2xy + 4y 3 z + 3x 2 z, ∂yf(x, y, z) = x 2 + 12xy 2 z, ∂zf(x, y, z) = −3z 2 + 4xy 3 + x 3 . ∂zf(1, 2, −1) = −3+32+1 = 30 = 0 z = ϕ(x, y) (1, 2) ϕ(1, 2) = −1 C 1 f ∈ C 1 ∂xϕ(x, y) = − ∂xf(x, y, ϕ(x, y)) ∂zf(x, y, ϕ(x, y)) = −2xy + 4y3ϕ(x, y) + 3x2ϕ(x, y) −3ϕ2 (x, y) + 4xy3 + x3 ∂yϕ(x, y) = − ∂xf(x, y, ϕ(x, y)) ∂zf(x, y, ϕ(x, y)) = − x2 + 12xy 2 ϕ(x, y) −3ϕ 2 (x, y) + 4xy 3 + x 3 ∂xϕ(1, 2) = 31/30 ∂yϕ(1, 2) = 47/30 f1(x, y, z) = x 2 + y 2 − z 2 + 2 = 0, f2(x, y, z) = xy + 2yz − xz + 7 = 0,
x = 1 y = −1 z = 2 y z x x = 1 y(x) z(x) x = 1 −1 2 y ′ (1) z ′ (1) F (x, y, z) := (f1(x, y, z), f2(x, y, z)). F : R 3 → R 2 F Jac(F )(x, y, z) = 2x 2y −2z y − z x + 2z 2y − x y z x y z (1, −1, 2) Jac(F ) f1, f2 y z Jac(F )(1, −1, 2) = 2 −2 −4 −3 5 −3 . −2 −4 , Jacy,z(F )(1, −1, 2) = 5 −3 (1, −1, 2) det Jacy,z(F )(1, −1, 2) = 26 = 0 x = 1 y z y = y(x) z = z(y) y(1) = −1 z(1) = 2 ϕ(x) = (y(x), z(x)) Jacy,z(F )(1, −1, 2) [Jacy,z(F )(1, −1, 2)] −1 = 1 26 −3 4 −5 −2 x F (1, −1, 2) F Jacx(F )(1, −1, 2) = y ′ (1) = 9/<strong>13</strong> z ′ (1) = 2/<strong>13</strong> 2 −3 . ϕ ′ y ′ (x) (1) = z ′ = −[Jacy,z(F )(1, −1, 2)] (x) −1 Jacx(F )(1, −1, 2) = − 1 −3 4 2 = − 26 −5 −2 −3 1 −18 26 −4 1 p1 > 0 p2 > 0 m3 m1 m2 1/3 1/2 m3 = m 1/3 1 m1/2 2 . p1 > 0 p2 > 0 (m1, m2) p1 p2 (m1, m2) m3 1 pi mi i = 1, 2 I(m1, m2, m3, p1, p2) = m3 − p1m1 − p2m2, .
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
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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: R 2 D ⊆ R 2 f : D → R (x0,
Page 85: ⎛ 1 ⎜ 3 F (m1, m2, p1, p2) =
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
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I := F · ˆn dσ ˆn Σ =
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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
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˙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
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y ′ + y 1 x = tan x sin x y ′
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λ 2 − 4λ − 5 = 0, A
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F (x, y, ˙y) = 0, F : A → R A
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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(
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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 = −
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x = cos θ y = sin θ F F (co
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I := arctan y dx − xy dy γ γ
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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)
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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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