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

dσ = det 2 B1 + det 2 B2 + det 2 B3 du dv Jac ϕ dσ = |∂uϕ(u, v) ∧ ∂vϕ(u, v)| du dv (x0, y0, z0) (u0, v0) ∈ I×J ϕ(u0, v0) = (x0, y0, z0) ˆn(x0, y0, z0) = ∂uϕ(u0, v0) ∧ ∂vϕ(u0, v0) |∂uϕ(u0, v0) ∧ ∂vϕ(u0, v0)| (u0, v0) v v/|v| Σ ϕ ˆn(x, y, z) (u, v) ∈ I × J ˆn ◦ ϕ(u, v) · ∂uϕ(u, v) ˆn ◦ ϕ(u, v) · ∂vϕ(u, v) det ⎛ ⎝ n1 ◦ ϕ(u, v) ∂uϕ1(u, v) ∂vϕ1(u, v) n2 ◦ ϕ(u, v) ∂uϕ2(u, v) ∂vϕ2(u, v) n3 ◦ ϕ(u, v) ∂uϕ3(u, v) ∂vϕ3(u, v) (u, v) Σ (ū, ¯v) P (¯x, ¯y, ¯z) Σ (ū, ¯v) ϕ(ū, ¯v) = P (¯x, ¯y, ¯x) (ū, ¯v) Σ f(x, y, z) = 0 ∇f(x, y, z) Σ Σ Σ ∇f Σ ϕ f = 0 ∇f ∂uϕ ∧ ∂vϕ F ⎛ F · ˆn dσ = det ⎝ F1 ⎞ ◦ ϕ ∂uϕ1 ∂vϕ1 F2 ◦ ϕ ∂uϕ2 ∂vϕ2 ⎠ du dv S I J F3 ◦ ϕ ∂uϕ3 ∂vϕ3 γ G γ : [0, T ] → R 3 C 1 γ G · ds = T 0 G(γ(t)) · ˙γ(t) dt. Σ ψ : [a, b]×[c, d] → R 3 γ1(t) = ψ(t, c) t ∈ [a, b] γ2(t) = ψ(b, t) t ∈ [c, d] γ3(t) = ψ(b + a − t, d) t ∈ [a, b] γ4(t) = ψ(a, d + c − t) t ∈ [a, b] ψ [a, b] × [c, d] ψ(a, t) = ψ(b, t) t ∈]c, d[ ψ(t, c) = ψ(t, d) t ∈]a, b[ ⎞ ⎠ .
ψ(a, t) = ψ(b, t) t ∈]c, d[ ψ(t, c) = ψ(t, d) t ∈]a, b[ γ1 γ3 ψ(a, t) = ψ(b, t) t ∈]c, d[ ψ(t, c) = ψ(t, d) t ∈]a, b[ γ2 γ4 ψ(a, t) = ψ(b, t) t ∈]c, d[ ψ(t, c) = ψ(t, d) t ∈]a, b[ F Σ F Σ rot F · ˆn dσ = F · ds. Σ F R3 Ω C F F · ˆn dσ = div F (x, y, z) dx dy dz. C Ω Ω S div F = 0 S γ Π Π Σ Σ ∩ S = γ Σ S S ∪ Σ = C Ω F · ˆn dσ + F · ˆn dσ = F · ˆn dσ = div F (x, y, z) dx dy dz = 0, S Σ C Σ Π F Σ F Σ Σ Π f : R → R T f(x) = f(x + T ) x ∈ R a ∈ R T 0 f(x) dx = T/2 −T/2 f(x) dx = γ Ω a+T f : R → R f(x) = f(−x) a f(x) dx a g : R → R g(−x) = −g(x) f(x) dx = 2 −a a −a a 0 g(x) dx = 0 f(x) dx
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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 →
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
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I =]a, b[ R {fn}n∈N fn : I
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K [0, +∞[ R > 0 B(0, R) ⊇
Page 38 and 39:
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
Page 51 and 52:
(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
Page 60 and 61:
∂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
Page 74 and 75:
⎧ ⎪⎨ 1 + λ(2x − 2) = 0, 1
Page 76 and 77:
f(x, y) = x 2 + y 2 V = {(x, y)
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⎧⎪ 2x + λ(2x + y) + 2µx = 0
Page 81 and 82:
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,
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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
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: ϕ : I ×J → R 3 I, J R
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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