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γ lunghezza(γ) = |r ′ (θ)| dθ = 3 I 5π 0 3 1 + θ2 dθ = 5π 2 1 + <strong>25</strong>π2 + log(5π + 1 + <strong>25</strong>π2 ) , √ 1 + t 2 µ(x(θ), y(θ)) = µ(ρ(θ) cos θ, ρ(θ) sin θ) = ρ 2 (θ) = 9θ 2 . massa(γ) = µ(x(θ), y(θ)) ˙x 2 (θ) + ˙y 2 (θ) dθ = 27 I 5π 0 θ 2 1 + θ 2 dθ. w = 2θ 2 + 1 √ w 2 − 1 5π 50 2 π+1 massa(γ) = 27 θ 0 2 1 + θ2 dθ = 27 w2 − 1 dw 8 1 = 27 1 2 + 50π (1 + 50π 16 2 ) 2 − 1 − log 1 + 50π 2 + (1 + 50π2 ) 2 − 1 . T r z a a > r C r zy (0, a, 0) C := {(0, s cos φ + a, s sin φ) : 0 ≤ s ≤ r, φ ∈ [0, 2π]} θ ∈ [0, 2π] (x, y, z) z θ ( x 2 + y 2 cos θ, x 2 + y 2 sin θ, z) T := {(|s cos φ + a| cos θ, |s cos φ + a| sin θ, s sin φ) : ρ, φ ∈ [0, 2π], 0 ≤ s ≤ r} a > r s cos φ + a > −s + a > −r + a > 0 T := {((s cos φ + a) cos θ, (s cos φ + a) sin θ, s sin φ) : ρ, φ ∈ [0, 2π], 0 ≤ s ≤ r}, T ϕ(s, θ, ϕ) = ((s cos φ + a) cos θ, (s cos φ + a) sin θ, s sin φ). T V = T dx dy dz. ⎛ cos φ cos θ −(s cos φ + a) sin θ Jac(ϕ)(s, θ, ϕ) = ⎝ cos φ sin θ (s cos φ + a) cos θ ⎞ −s sin φ cos θ −s sin φ sin θ ⎠ . sin φ 0 s cos φ det Jac(ϕ)(s, θ, ϕ) = sin φ(s(s cos φ + a) sin 2 θ sin φ + s sin φ cos 2 θ(s cos φ + a))+ + s cos φ(cos φ cos 2 θ(s cos φ + a) + (s cos φ + a) sin 2 θ cos φ) = s sin 2 φ(s cos φ + a) + s cos 2 φ(s cos φ + a) = s(s cos φ + a) > 0
V = 2π 2π r s(s cos φ + a) dr dφ dθ 0 0 0 r 2π = 2πa (s 0 0 2 cos φ + as) dφ ds = 4π 2 r a s dr = 2π 0 2 ar 2 . C πr2 2πa V = 2π2ar2 I = cos(x + y)e x−y dxdy D D = {(x, y) ∈ R 2 : |x + y| ≤ π , |x − y| ≤ 1}. 2 u = x+y v = x−y x = (u+v)/2 y = (u−v)/2 −1/2 1/2 I = 1 1 π/2 cos u e 2 −1 −π/2 v du dv = e − e −1 = 2 sinh(1). 1 + y I = x dx dy, D 1 − y D y 4 + x 2 − 2x = 0 1 − y 4 = (x − 1) 2 (1 − y)(1 + y)(1 + y 2 ) = (x − 1) 2 −1 ≤ y ≤ 1 1 − 1 − y4 < x < 1 + 1 − y4 1 √ 1+ 1−y4 1 √ 1+ 1−y4 1 + y I = x dx dy = 1 − y = 1 2 = 2 = 4 −1 1− √ 1−y4 1 −1 1 −1 1 0 −1 1− √ 1−y 4 1 + y x dx dy 1 − y 1 + y 1 − y ((1 + 1 − y4 ) 2 − (1 − 1 − y4 ) 2 1 1 + y ) dy = 2 1 − y4 dy −1 1 − y (1 + y) 1 + y2 1 dy = 2 1 + y2 dy − 2 y 1 + y2 dy −1 1 + y 2 dy = 2( √ 2 + arc sinh(1)) = 2( √ 2 + log(1 + √ 2)) |(x, y)| −p B = B(0, 1) ⊆ R 2 p < 2 R 2 \ B(0, 1) p > 2 |(x, y, z)| −p B(0, 1) ⊆ R 3 p < 3 R 3 \ B(0, 1) p > 3 p = 2 |(x, y)| B −p 2π 1 1 dx dy = ρ dρ dθ = 2π 0 0 ρp p < 2 p = 2 2π 0 1 ρ = 2π[log ρ]10 = +∞, B(0, 1) ⊆ R2 p < 2 1 1 0 ρ 1−p dρ = 2π (2 − p) [ρ2−p ] ρ=1 ρ=0 ,
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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 =
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 and 84: x = 1 y = −1 z = 2 y z
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 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) = (
Page 136 and 137: C = {(x, y, z) : (x, y) ∈ D, λ <
Page 138 and 139: ˆn = ∇G = ((∇G)1, (∇G)2, (
Page 140 and 141: 2π 0 cos θ sin θ dθ = 1 2 2π
Page 142 and 143: L − F · ˆn dσ = − L− √
Page 145 and 146: K ⊆ R n χK : R n → {0, 1}
Page 147 and 148: f [1, +∞[ [0, 1] [0
Page 149 and 150: f α k = 1 x α + k α , sα n = n
Page 151 and 152: D R X n 1 1 X C ℓ
Page 153 and 154: 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
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Jac ϕ(θ, y) = ⎛ ⎜ ⎝ − y
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z = (x, y) ˙z = Az + B(t)
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Q0 Q2 Q1 Q1
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˙x = t − x 2 x(0) = a a ∈ R
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2π xt sin t F (x) = e dt x ∈ R
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xz 3 dx dy dz, T T y
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α > 0 lim (x,y)→(0,0) (x,y)=(0,0
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F (0, 1) = 0 ∂yF (x, y) = 3y 2
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(u + v) D 2 + (u − v) 2 1 + 2(u2
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α = = 2 3 π. 2π 0
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F D z = 0 (0,
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y ′ = 2xy − x y(0) = 3/2
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Γ f(x, y) = 0 f : R 2 → R
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∇f(x0, y0) = (0, 0) (x0, y0)
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ϕ : I ×J → R 3 I, J R
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ψ(a, t) = ψ(b, t) t ∈]c, d[
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[
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Y K Ω R × Ω I R f :
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1 ω(x, y) = M(x, y) dx + N(x, y)
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M(x, y) dx + N(x, y) dy = 0 1 M, N
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x, y z dy dz
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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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