pdf (it, 2.898,97 KB, 25/01/13)

More documents

Recommendations

Info

$(Microsoft PowerPoint - 03_Gruppi_Alcani [modalit\340 compatibilit ...$

z = (x, y) ˙z = Az + B(t) −2 A = −5 −3 , −1 3e−2 B(t) = 0 . T = tr(A) = −3 D = det(A) = −13 λ2 − T λ + D = 0 λ2 + 3λ − 13 = 0 λ1 = 1 −3 − 2 √ 61 , λ2 = 1 −3 + 2 √ 61 . D = 0 (0, 0) −3y = ˙x + 2x − 3e −2t ˙y = −5x − y −3 ˙y = ¨x + 2 ˙x + 6e−2t ˙y −3(−5x − y) = ¨x + 2 ˙x + 6e −2t . ¨x + 2 ˙x − 15x − 3y + 6e −2t = 0 y ¨x + 2 ˙x − 15x + ( ˙x + 2x − 3e −2t ) + 6e −2t = 0. x ¨x + 2 ˙x − 15x + ˙x + 2x − 3e −2t + 6e −2t = 0. ¨x − (−2 − 1) ˙x + (2 − 15)x − 3e −2t + 6e −2t = 0 ¨x − T ˙x + D x = −3e −2t c1e λ1t + c2e λ2t −2 qe −2t q ∈ R e −2t 4q − 6q − 13q = −3 q = 1/5 x(t) = c1e λ1t + c2e λ2t + 1 5 e−2t ˙x(t) = c1λ1e λ1t + c2λ2e λ2t − 2 5 e−2t . y(t) = − 1 3 c1λ1e λ1t + c2λ2e λ2t − 2 = 1 1 e− 2(3+ 6 √ 61)t − ⎧ 1 ⎨ − x(t) = c1e c1, c2 ∈ R ⎩y(t) = 1 6 1 + √ 61 5 e−2t + 2 c1e √ 61t + 2(3− √ 1 61)t − + c2e 2(3+ √ 61)t e + −2t 5 1 e− 2(3+ √ 61)t c1e λ1t + c2e λ2t 1 + 5 e−2t √ 61 − 1 c2 + 6e 1 2( √ 61−1)t . − 3e −2t − 1 + √ 61 c1e √ √ 61t + 61 − 1 c2 + 6e 1 2( √ 61−1)t a ∈ R x ′ (t) = t − x 2 (t), x(0) = a,
a (−∞, 0] a [0, +∞) ˙x = 0 R := {(t, x(t)) : ˙x(t) > 0} = {(t, x) : t < x 2 }, ˙t = 1 ˙x = t − x 2 R (˙t, ˙x) (t,x)∈R = (1, 0) R ˙x x → +∞ x ′′ = 1 − 2xx ′ = 1 − 2x(t − x 2 ) γ := {(t, x) : x ′′ = 0} = {(t, x) : 1 − 2xt + 2x 3 } = {(t, x) : t = 1/(2x) + x 2 }. γ t = t(x) = 1/(2x) + x 2 t = x 2 x → +∞ x → 0 ± ±∞ x > 0 R x − 3 1/2 x x < 0 R 2 \ γ Q0 := {(t, x) : x ′′ < 0, x > 0, t > 0} = {(t, x) : 1 − 2xt + 2x 3 < 0 x > 0, t > 0} Q1 := {(t, x) : x ′′ > 0} = {(t, x) : 1 − 2xt + 2x 3 > 0} Q2 := {(t, x) : x ′′ < 0, x < 0} = {(t, x) : 1 − 2xt + 2x 3 < 0, x < 0}. γ F (t, x) = 0 F (t, x) = 1 − 2xt + 2x 3 (t, x) ∈ γ ˆn(t, x) = ±∇F (t, x) = ∓(2x, 2t − 6x 2 ) Q0 Q1 Q2 Q0 Q0 ∂Q0 x > 0 Q0 ˆn0(t, x) = (2x, 2t − 6x 2 ) Q2 Q2 ∂Q2 x < 0 Q2 ˆn2(t, x) = (2x, 2t − 6x 2 ) = ˆn0(t, x) Q1 ˆn1(t, x) = −n0(t, x) = −(2x, 2t − 6x 2 ) ˆni · (1, ˙x) ∂Qi ˆn0 · (1, ˙x) t=1/(2x)+x 2 = [2x + (t − x 2 )(2t − 6x 2 )] t=1/(2x)+x 2 = 2x + 1 2x = 2x + 1 1 3 + − 2x2 x x 1 = > 0. 2x2 ˆn2 · (1, ˙x) t=1/(2x)+x2 = ˆn0 · (1, ˙x) t=1/(2x)+x2 = 1 > 0. 2x2 ˆn1 · (1, ˙x) t=1/(2x)+x2 = −ˆn0 · (1, ˙x) t=1/(2x)+x2 = − 1 < 0. 2x2 1 x + 2x2 − 6x 2
Page 3 and 4:
R R n
Page 5:
2 × 2 SO(
Page 9 and 10:
R a
Page 11:
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 : [
Page 21:
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 >
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 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 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 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,
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 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 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
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
show all

pdf (it, 2.898,97 KB, 25/01/13)

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?