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t ∈ I LK > 0 f(t, y1) − f(t, y2)Y ≤ LKy1 − y2Y t ∈ K K ⊆ I y1, y2 ∈ Y t0 ∈ I y0 ∈ Y ϕ ∈ C 1 (I, Y ) ϕ ′ (t) = f(t, ϕ(t)) I ϕ(t0) = y0 Y K I R K I f : I × Y → Y K × Y ∂Y f(t, y) L(Y ) ≤ LK < +∞ (t, y) ∈ K × Y Y K n ∂y1 f(t, y), ..., ∂ynf(t, y) K × Y Y K Ω R × Y f : Ω → Y (t0, y0) ∈ Ω L, δ0, r0 > 0 B(t0, δ0] × B(y0, r0] ⊆ Ω f(t, y1) − f(t, y2)Y ≤ Ly1 − y2Y (t, y) ∈ B(t0, δ0] × B(y0, r0] (t0, y0) ∈ Ω δ > 0 ϕ ∈ C 1 (B(t0, δ], Y ) y ′ = f(t, y) y(t0) = y0 ψ ∈ C 1 (B(t0, δ], Y ) t0 ϕ(t) = ψ(t) t0 Y K Ω R×Y f : Ω → Y f ∂Y f(t, y) Ω Y Kn ∂ykf(t, y) k = 1, ..., n Ω y ′ = f(t, y) I R φ, ψ : I → Y y ′ = f(t, y) I y ′ = f(t, y) y(t0) = y0 Φ(t, t0, y0) t y(t0) = y0 ˙y = f(t, y) y(t0) = y0 (t0, y0) Φ D ⊃ I × I × Ω I t0 Ω y0 Φ : D → Y f t t ↦→ y(t) t ↦→ y(t + c) Φ(t, y0) = φt(y0) t y(0) = y0 φ0(y0) = y0 φs ◦ φt(y0) = φs+t(y0) Y = R n y = f(y) Y n = 1, 2, 3
Y K Ω R × Ω I R f : Ω → Y ϕ : I → Y β = sup I α = inf I c ∈ I ϕ ′ (t) [c, β[ ]α, c] β = +∞ α = −∞ lim t→β − ϕ(t) = yβ lim t→α + ϕ(t) = yα Y (β, yβ) /∈ Ω (β, yβ) /∈ Ω K Ω U a = inf I V b = sup I t ∈ U ∪ V (t, ϕ(t)) /∈ K Ω K Ω δ = δ(K) > 0 K f (t0, y0) ∈ Ω [t0 − δ, t0 + δ] A Y g : A → Y ϕ : I → A y ′ = g(y) C A b = sup I V b ϕ(t) /∈ C t ∈ V ϕ C b = +∞ a = inf I y ′ = g(y) g : A → Y A Y E ∈ C 1 (A, R) ϕ : I → A E ◦ ϕ I R µ : I × [0, +∞[→ [0, +∞[ Y ϕ : I → Y u : I → [0, +∞[ ϕ(t0)Y ≤ u(t0) ϕ ′ (t)Y < µ(t, ϕ(t)Y ) µ(t, u(t)) ≤ u ′ (t) t ≥ t0 t ∈ I t ϕ(t)Y ≤ u(t) ϕ ′ (t)Y < µ(t, ϕ(t)Y ) µ(t, u(t)) ≤ −u ′ (t) t ≤ t0 t ∈ I t ϕ(t)Y ≤ u(t) t = t0 ϕ(t)Y < u(t) ϕ ′ (t)Y ≤ µ(t, ϕ(t)Y ) µ(t, u(t)) < u ′ (t) ϕ ′ (t)Y ≤ µ(t, ϕ(t)Y ) µ(t, u(t)) < −u ′ (t) Ω R × R f : Ω → R I R t ↦→ y(t) t ↦→ u(t) I t0 ∈ I y(t0) ≤ u(t0) t > t0 t ∈ I y ′ (t) ≤ f(t, y(t)) f(t, u(t)) ≤ u ′ (t) t ≥ t0 y(t) ≤ u(t) t ∈ I t ≥ t0 t0 I R f, g : I × R → R x : I → R y : I → R ˙x ≤ f(t, x(t)) ˙y ≥ g(t, y(t)) t ∈ I f(t, x(t)) ≤ g(t, y(t)) t ∈ I x(t0) ≤ y(t0) x(t) ≤ y(t) t ∈ I t ≥ t0 x(t0) ≥ y(t0) x(t) ≥ y(t) t ∈ I t ≤ t0 a > 0 x : [a, +∞[→ R limt→+∞ x(t) ∈ R limt→+∞ ˙x(t) = γ ∈ R ∪ {±∞} γ = 0 I R to ∈ I Y
Page 3 and 4:
R R n
Page 5:
2 × 2 SO(
Page 9 and 10:
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, ...
Page 17 and 18:
(X, τ) D ⊆ X X
Page 19 and 20:
f : D → R D ⊆ Rn γ : [
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A =]0, +∞[×]0, +∞[ f : A →
Page 24 and 25:
lim (x,y)→(0,0) (x,y)=(0,0) |xy|
Page 26 and 27:
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 >
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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 =
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X, Y D ⊆ X f : D → Y u ∈
Page 55 and 56:
H A 2 × 2 λ 2
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(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) =
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α ∈ R (0, 0) f(x, y) = 2
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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,
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x = 1 y = −1 z = 2 y z
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⎛ 1 ⎜ 3 F (m1, m2, p1, p2) =
Page 88 and 89:
k = 3√ 4 ξ ζ
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x = ρ cos θ y = ρ sin θ ρ 2 (
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f(x, y, z) = ez+x2 f + αx + y
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ϕ(1) = 0 ϕ ′ (1) = − ∂xf(1
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y 3 − xy 2 + x 2 y = x − x 3
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X = {(ρ, θ) ∈ R 2 : ρ > 0, 0 <
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arcsin sin θ = θ θ ∈] − π
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I R r : I → Rd γ µ : r
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γ lunghezza(γ) = |r ′ (θ)|
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p = 2 |(x, y)| −p dx dy = R 2 \B
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ρ 3 cos θ = aρ 2 (cos 2 θ − s
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Γ := {(x, y) ∈ R 2 : (x 2 + y 2
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x Γ Q2 Q3 x Γ \
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(x 2 + y 2 )(y 2 + x(x + 1)) = 4xy
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4 ±1, ±2, ±4 ±1, ±2 4
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−2 + 8t − 4t 2 t = 1/2(2
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a ∈ R R 2 (x 2 + y 2 ) 3 = 4
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(x 2 + y 2 ) 3 = 4x 2 y 2
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a > 0 x = at at2 , y = . 1
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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, (
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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,
Page 193 and 194: e |uk(x, t)| = C −t sin( 4k(1
Page 195: ∞ n=1 bne −x sin nx = x(
Page 198 and 199: T ˙ Tn(t) = −5n 2 Tn(t) Tn(
Page 200 and 201: ⎧ −utt + 3uxx = 0 ]0, π[×]0,
Page 202 and 203: ∆ = 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
Page 212 and 213: 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[
Page 243: [
Page 247 and 248: 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
Page 251: 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
Page 271: 2 × 2 t ↦→ x(
Page 274 and 275: ∆ < 0 α = −d 2c ω = √ |
Page 276 and 277: 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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