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

v = (v1, ..., vm) df(p)(v) = Jac f(p)(v) = ⎛ ⎜ ⎝ ∂x1f1(p) . . . ∂xnf1(p) ∂xnfm(p) . . . ∂xnfm(p) ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ f : R n → R f Jac f(p) = (∂x1 f(p), . . . ∂xnf(p)) R n n ∇f(p) grad f(p) f p f p f p Df(p)u = ∂uf(p) X = Rn Y = R p Rn R h = (h1, ..., hn) Rn h = n j=1 hjej ⎛ ⎞ n n n df(p)(h) = df(p) ⎝ ⎠ = df(p)(ej) · hj = ∂xjf(p) hj ∈ R. j=1 hjej df(p) = j=1 n j=1 ∂xjf(p) dxj, df (p) h = (h1, ..., hn) n ∂xjf(p) hj. j=1 D R n p ∈ D f : D → Y f p f p D(αf + βg) = αDf + βDg. D(f ◦ g)(p) = Df(g(p)) ◦ Dg(p) ◦ C 1 f : D → Y D R n C 1 (D, Y ) D f X, Y D ⊆ X f : D → Y u ∈ X x ∈ D ∂uf(x) ∂uf : D → Y x Y ∂uf(x) v ∈ X ∂v(∂uf)(x) f D f ′ : D → L(X, Y ) f ′ p f p f ′′ (p) D 2 f(p) f ′′ (p) ∈ L(X, L(X, Y )) L 2 (X ×X, Y ) L : X × X → Y K R C E a, b ∈ R f : E × [a, b] → K F (x) := F : E → K b a f(x, t) dt j=1 v1 vm ⎞ ⎟ ⎠ .
X K E X a, b ∈ R f : E × [a, b] → K u X x ∈ E t ∈ [a, b] ∂uf(x, t) E × [a, b] ∂uf(x, t) E ∂uF (x) = b a ∂uf(x, t) dt X E X I R f : E × I → Y Y Φ : E × I × I → Y Ψ(x, α, β) = β α f(x, t) dt Ψ Ψ α, β ∂βΨ(x, α, β) = f(x, β), ∂αΨ(x, α, β) = −f(x, α); X ≈ Kn ∂if(x, t) i = 1...n Ψ(x, α, β) α, β Ψ ′ n β (x, α, β)(h, △α, △β) = ∂jf(x, t) dt hj + f(x, β)△β − f(x, α)△α, j=1 α h = (h1, ..., hn) ∈ Kn x ↦→ α(x) x → β(x) R I α, β : E → I ⊆ R ∂kG(x) = β(x) α(x) G(x) = β(x) α(x) f(x, t) dt ∂kf(x, t) dt + f(x, β(x))∂kβ(x) − f(x, α(x))∂kα(x). f : R 2 → R f(x, y) = x 2 sin y f(x, y) = |x| f(x, y) = |xy| f(x, y) = |x| + |y| f(x, y) = |xy| f(x, y) = sign(2 − x 2 − y 2 ) |2 − x 2 − y 2 | ∂xf(x, y) = 2x sin y ∂yf(x, y) = x 2 cos y R 2 R 2 Df(x, y) = 2x sin y dx + x 2 cos y dy. ∂xf(x, y) = sign(x) 2 |x| x = 0 ∂yf(x, y) = 0 R 2 \ ({0} × R) Df(x, y) = sign(x) 2 |x| dx ∂xf(x, y) = |y| sign(x) ∂yf(x, y) = |x| sign(y) xy = 0 Df(x, y) = |y|sign(x) dx + |x|sign(y) dy ∂xf(x, y) = sign(x) ∂yf(x, y) = sign(y) xy = 0 Df(x, y) = sign(x) dx + sign(y) dy
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: X Y T : X → Y T ℓ >
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)|
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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
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x Γ Q2 Q3 x Γ \
Page 115 and 116:
(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
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 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
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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