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S ψ(x, y) = (ψ1, ψ2, ψ3) = (x, y, f(x, y)) = (x, y, xy/2) dσ = ⎛ Jac ψ(x, y) = ⎝ 1 0 0 1 y/2 x/2 2 2 det 2 1 0 0 1 + det 2 1 0 y/2 x/2 ⎞ ⎠ + det 2 0 1 y/2 x/2 dx dy = 1 + x2 y2 + dx dy. 4 4 Jac ψ dσ = ∂xψ ∧ ∂yψ = − y , −x , 1 dx dy = 1 + 2 2 x2 4 + y2 4 Area(S) = S dσ = D 1 + x2 4 + y2 4 dx dy. 1 dxdy = 4 + y2 + x2 dxdy. 2 D x = ρ cos θ y = ρ sin θ ρ Area(S) = = π 2 2π 0 16 4 2 √ 3 0 √ t dt = π 2 1 4 + ρ2 ρ dρ dθ = π 2 t 3/2 3/2 t=16 t=4 2 √ 3 0 = π 56π (64 − 8) = 3 3 . 4 + ρ 2 ρ dρ = π 2 12 0 √ 4 + w dw S G(x, y, z) = z−f(x, y) = z−xy/2 G(x, y, z) = 0 ± ∇G (−y/2, −x/2, 1) (−y, −x, 2) = ± = ± , |∇G| |(−y/2, −x/2, 1)| x2 + y2 + 4 ˆn(x, y, z) = (n1, n2, n3) = (−y, −x, 2) x 2 + y 2 + 4 ,
⎛ det ⎝ n1 n2 n3 ∂xψ1 ∂xψ2 ∂xψ3 ∂yψ1 ∂yψ2 ∂yψ3 ⎛ −y ⎞ √ 1 0 x2 +y2 +4 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ −x ⎠ = det ⎜ √ ⎟ 0 1 ⎟ ⎜ x2 +y2 +4 ⎟ ⎜ ⎟ ⎝ ⎠ √ 2 y/2 x/2 x2 +y2 +4 ⎛ ⎞ −y 1 0 1 = det ⎝ −x 0 1 ⎠ = x2 + y2 + 4 2 y/2 x/2 2 + y2 /2 + x2 /2 x2 + y2 + 4 = 1 x2 + y2 + 4 > 0. 2 F = (F1, F2, F3) Φ(S, ⎛ F ) = det ⎝ D F1 ⎞ ◦ ψ ∂xψ1 ∂yψ1 F2 ◦ ψ ∂xψ2 ∂yψ2 ⎠ dxdy = ⎛ x det ⎝ y 1 0 0 1 ⎞ ⎠ dxdy F3 ◦ ψ ∂xψ3 ∂yψ3 = (1 − xy) dxdy = Area(D) − D 1 y/2 x/2 2π √ 2 3 xy dxdy = 12π − r D D 0 0 2 cos θ sin θ ρ dρ dθ √ 2 3 = 12π − r 0 3 2π dr · cos θ sin θ = 12π. 12π 0 λ ∈ R λ < min{f(x, y) : x, y ∈ D} λ f D λ < 0 div F (x, y, z) = ∂F1 ∂x + ∂F2 ∂y + ∂F3 ∂z C S S − := D × {λ} C (0, 0, −1) C {(x, y, λ) : x, y, ∈ D} S = 2. L = {(x, y, z) : x 2 + y 2 = 12, λ ≤ z ≤ f(x, y)}. (x, y, 0)/ x2 + y2 div F dxdydz = F · ˆn dσ. C ∂C 2Volume(C) = Φ(S, F ) + 2 f(x,y) dz dxdy = Φ(S, D λ F ) − S − D F · ˆn dσ + dσ + L L F · ˆn dσ x 2 + y 2 x 2 + y 2 dσ 2Volume(C) = Φ(S, F ) − Area(D) + √ 12 Area(L)
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,
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 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 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,
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
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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
Page 219 and 220:
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
Page 235 and 236:
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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