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I := arctan y dx − xy dy γ γ T (1, 0) (0, 1) (0, −1) T M := x 2 dx dy. T T = {(x, y) ∈ R 2 : −1 ≤ y ≤ 1, 0 ≤ x ≤ 1 − |y|} γ = {(t, t − 1) : t ∈ [0, 1]} ∪ {(1 − t, t) : t ∈ [0, 1]} ∪ {(−t, 0) : t ∈ [1, −1]} P (x, y) dx + Q(x, y) dy = arctan y dx − xy dy, T I = − (∂xQ − ∂yP ) dxdy = − y − T T 1 1 + y2 dx dy 1 1−|y| = − y − 1 1 + y2 dx dy = − = 2 −1 1 −1 1 0 0 y(1 − |y|) dy + 1 − y 1 + y 2 1 −1 1 − |y| dy 1 + y2 = [2 arctan y − log(1 + y 2 )] y=1 π y=0 = M := x 2 dx dy = Q(x, y) = x3 /3 P = 0 x M = Q(x, y) dy = γ γ 3 dy = 3 = 1 1 1 + = 12 12 6 . M = x 2 dx dy = = 1 3 = 2 3 T T 1 −1 1 0 1 0 1 (1 − |y|) 3 dy −1 T t 3 3 (1 − y) 3 dy = 1 6 . − log 2. 2 (∂xQ − ∂yP ) dxdy, 1 (1 − t) dt + 0 3 1 dt − 0 dt 3 −1 1−|y| x 2 dx dy Γ = {(x, y) ∈ R 2 : (x 2 + y 2 + 12x + 9) 2 = 4(2x + 3) 3 }, 0
cos(3θ) = cos θ(1 − 4 sin 2 θ) = cos θ(cos 2 θ − 3 sin 2 θ) Γ Γ 2π 3 Γ P1 = (−1, 0) P2 = (1/2, √ 3/2) P3 = (1/2, − √ 3/2) Γ P0 y = ϕ1(x) ϕ1(−1) = 0 ϕ ′ 1 (0) √ Γ P1 y = ϕ2(x) ϕ2(1/2) = 3/2 ϕ ′ 2 (1/2) Γ P2 y = ϕ3(x) ϕ3(1/2) = − √ 3/2 ϕ ′ 3 (1/2) h(x, y) = x2 + y2 Γ Γ (3, 0) Γ F (x, y) = (x 2 + y 2 + 12x + 9) 2 − 4(2x + 3) 3 . F (x, y) = F (x, −y) cos 2 θ + sin 2 θ = 1 cos 3θ = cos(θ + 2θ) = cos θ cos 2θ − 2 sin 2 θ cos θ = cos θ(1 − 2 sin 2 θ) − 2 sin 2 θ cos θ = cos θ(1 − 4 sin 2 θ) F (ρ cos θ, ρ sin θ) = ρ 4 (sin 4 (θ) + cos 4 (θ) + 2 sin 2 (θ) cos 2 (θ))+ + ρ 3 (−8 cos 3 (θ) + 24 sin 2 (θ) cos(θ)) + 18ρ 2 − 27 = ρ 4 (sin 2 (θ) + cos 2 (θ)) 2 − 8ρ 3 cos θ(cos 2 (θ) − 3 sin 2 (θ)) + 18ρ 2 − 27 = ρ 4 − 8ρ 3 cos(3θ) + 18ρ 2 − 27 θ 2π/3 2π/3 (0, 0) /∈ Γ f(ρ) := ρ4 + 18ρ 2 − 27 8ρ 3 = cos(3θ) Γ Γ (ρn, θn) ρn → +∞ lim sup +∞ n→∞ 2π/3 ∂yF (0, −1) = 0 ∂xF (0, −1) = 0 (0, −1) x = −1 α = ±2π/3 − 1 x ± 2 √ 3 y = −1 2 √ 3 P3 − √ 3 P2
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, +∞[)
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s ↦→ 1 − e −s s ≥ 0 1
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Fn : [0, +∞[→ R Fn(ρ) = 2n ρ
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I =]a, b[ R {fn}n∈N fn : I
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K [0, +∞[ R > 0 B(0, R) ⊇
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an = 0 n = 2k an = 4/(πn2 ) n
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5 1 18 = + 2 4 π2 ∞ n=0 ∞ n=0
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g(x) := x(π−x) x ∈ [0, π]
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X Y T : X → Y T ℓ >
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X K E X a, b ∈ R f : E × [a
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|f(x, y)| ≤ log(1 + 3y
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(x, y) ∈ R 2 xy =
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X, Y D ⊆ X f : D → Y u ∈
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H A 2 × 2 λ 2
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(2kπ, π/2+2hπ) (π+2kπ, 3π/2+2
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∂xf(x, y, z) = −2 cos x sin x =
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(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]
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x = cos θ y = sin θ θ ∈ [−π
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γ(t) = (t, 1/t) t → ±∞ L(x
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⎧ ⎪⎨ 1 + λ(2x − 2) = 0, 1
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f(x, y) = x 2 + y 2 V = {(x, y)
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⎧⎪ 2x + λ(2x + y) + 2µx = 0
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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) =
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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,
Page 155 and 156: C = −B ∂yu(x, y) = C/x 1 + y2 y
Page 157 and 158: dy N(x, y) = − dx M(x, y) N M
Page 159 and 160: 2xy dx + (x 2 + 1) dy = 0 (x 2 + y
Page 161 and 162: x − c = arcsin(2y − 1) 0 < y
Page 163: h(x, y) = 1/(|x| √ x2 − 1)
Page 166 and 167: ˙x = e t−x /x x(α) = 1 α ∈ R
Page 168 and 169: ˙x = x 2 /(1 − tx) x(0) = α α
Page 170 and 171: y ′′′ − 6y ′′ + 11y ′
Page 172 and 173: c ′ 1 (x) cos x + c′ 2 (x) sin
Page 175 and 176: y ′ + y 1 x = tan x sin x y ′
Page 177 and 178: 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
Page 183: y = xy ′ + f(y ′ ) F (x, y, p
Page 186 and 187: y → 0 −∞
Page 188 and 189: x > 0 x y = ± 1 + x x < 0
Page 191 and 192: [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 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 and 244: [
Page 245 and 246: Y K Ω R × Ω I R f :
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
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Φ y ′ = A(t) y ′ = A(t)
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det P = 4 √ 7 = 0 P −1 AP
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y ′′ (t) + py ′ (t) + qy(t) =
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y(0) = c1 + d1 = 1 ˙y(t) = (−c1
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a ′ x + b ′ y + c ′ = 0
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η2 g g(η) > 0 y0 < η <
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2 × 2 F ∈ R[x] n
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2 × 2 t ↦→ x(
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∆ < 0 α = −d 2c ω = √ |
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a = 0 b = 0 c = 0 ⎧ ⎪⎨ b
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SO(3) U(3) := {O ∈
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SO(3) Aξ = ω × ξ ξ
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x 2 + y 2 = 1 x 2 − y 2 = 1
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1 arcsin(x) + x − x2 dx = √ 1
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z 1 + cos z cos = ± 2 2 z 1
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