Esercizi svolti di analisi reale e complessa - Dipartimento di ...

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F ′ (t) = ∂ f ∂ x (g(t), 1 − g2 (t))g ′ ∂ f (t) + ∂ y (g(t), 1 − g2 (t))(−2g(t)g ′ (t)) . g(0) = 1 g ′ (t) = 1 cosh 2 t + t √ 1 + t2 g ′ (0) = 1 1) F ′ (0) = = ∂ f f (1, 0) − 2∂ (1, 0) = ∂ x ∂ y −2 . 0 ∂ ∂x = ∂ ∂z = f(x, h(x, z), z) = ∂f ∂x ∂ f (x, h(x, z), z), ∂ x f(x, h(x, z), z) = ∂f ∂z ∂ f (x, h(x, z), z), ∂ z (x, h(x, z), z) + ∂f ∂y ∂ f (x, h(x, z), z), ∂ y (x, h(x, z), z) + ∂f ∂y ∂ f (x, h(x, z), z), ∂ y ∂h (x, h(x, z), z) (x, z) = ∂x ∂ f (x, h(x, z), z) ∂ z ∂h (x, h(x, z), z) (x, z) = ∂z ∂ f (x, h(x, z), z) ∂ z ⎛ ⎞ 1 ⎜ ∂ h ⎟ · ⎝ (x, z) ⎠ ∂ x 0 ⎛ ⎜ · ⎝ y = f(x) C 2 x = 0 x 2 + sinh y + e xy = 1 , f(0) = 0 F (x) ≡ x 2 + sinh f(x) + e xf(x) − 1 ≡ 0 0 ∂ h (x, z) ∂ z 1 ⎞ ⎟ ⎠ .
f(0) = 0 0 = F ′ (0) = 2x + (cosh f(x))f ′ (x) + e xf(x) (f(x) + xf ′ (x)) = f ′ (0) x = 0 0 = F ′′ (0) = 2 + (sinh f(x))(f ′ (x)) 2 + (cosh f(x))f ′′ (x) + + e xf(x) (f(x) + xf ′ (x)) 2 + 2f ′ (x) + xf ′′ (x) = |x=0 = 2 + f ′′ (0) , f ′′ (0) = −2 ∂f ∂x = ∂ F ∂ y (0, 0) = [cosh y + xexy ] |(x,y)=(0,0) = ⎛ ⎜ ⎝ ∂f1 ∂x1 ∂f6 ∂x1 = 1 = 0. ∂f1 ∂x2 ∂f6 ∂x2 ⎞ ⎟ ⎠ ∂f ∂y = ⎛ ∂ ⎜ f(x, g(t)) = ⎜ ∂t ⎝ ⎛ ⎜ ⎝ ∂f1 ∂y1 ∂f6 ∂y1 ∂f1 ∂y2 ∂f6 ∂y2 3 ∂f1 j=1 ∂yj g′ j (t) 3 ∂f6 j=1 ∂yj g′ j (t) ⎞ ⎟ ⎠ ∂f1 ∂y3 ∂f1 ∂y3 ⎞ ⎟ ⎠ |x=0 =
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Rn Rn Rn
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Rn Rn Rn
Page 6 and 7: ℓ 1 ℓ ∞ x = {xn}n ∈ R N
Page 8 and 9: P ≡ {(x, y) ∈ R 2 y = x 2 (x
Page 10 and 11: ∂ 5 f (1, 1, .., 1) n > 5 ∂x1
Page 12 and 13: z = z(x, y) (1, 1) z 3 −
Page 14 and 15: K ⊂ R2 \ y = 1 2 0 < δ < β
Page 16 and 17: (i) ∞ n=0 n 3 2 n (ii) ∞ n=1 n
Page 18 and 19: v ∈ R n 1 f(x) ≡ x +
Page 20 and 21: X ⊂ R n o X = ∅ X Qx ,
Page 22 and 23: E = {(x, y) ∈ R 2 : x ≥ 0, (x 2
Page 24 and 25: R 3 Γ ≡ {(u(t), v(t)) : t ∈ (
Page 26 and 27: F F (x, y, z) = (x, y, z) ;
Page 28 and 29: Im {(1 + i) n + (1 − i) n } Re
Page 30 and 31: x dz σ 1 + i σ x dz
Page 32 and 33: f Imz ≥ 0 ∃ lim f(z) <
Page 34 and 35: ∞ n=2 1 − 1 n2 = 1 2 . a
Page 36 and 37: Ω ∂
Page 38 and 39: a) b) c) d) e) f) g) h) i) 2π 0
Page 40 and 41: Λ = {λ ∈ R n : λ = λ(t), a
Page 42 and 43: x, y ∈ E k > N0 |(fk(x) −
Page 44 and 45: x ∈ ℓ 1 k > N0 x1 ≤ x
Page 46 and 47: X
Page 48 and 49: limk→+∞ x (k) = limk→+∞ y (
Page 50 and 51: (⇐=) i = 1, . . . , m fi
Page 52 and 53: |f(x) − f(y)| ≤ 2 1 1
Page 54 and 55: L 4m3 m→+∞ −→ +∞ 3
Page 58 and 59: ∂f ∂y = ∂f ∂x = ∂f ∂t =
Page 60 and 61: • M1 = ( 2 √ 3 , 1 √ 3 ) M
Page 62 and 63: s(x, y) ≡ x2s(x) c(x, y) ≡ y2s
Page 64 and 65: f ∂xf(x, y) = 2x − y 2 = 0
Page 66 and 67: 2 3 4 4 i=1 xi i x2 4 i=1 xii x3 4
Page 68 and 69: dist lim dist (x(t), y(t)), y = t
Page 70 and 71: • y(t, β) = e βt .
Page 72 and 73: α > 0 • [0, 1) • [0, a
Page 74 and 75: N ≥ N0 |(1+x) α − N−1 k=0 |
Page 76 and 77: • [0, +∞) • [0, +∞)
Page 78 and 79: k 2k + 1 2n = (−1)k (2k + 1)! n=
Page 80 and 81: I − T f ′ (y)∞,∞ = max {|
Page 82 and 83: g Br(0) r ≡ ρ ρ = 2 In 2 .
Page 84 and 85: g(x, y) = e x2 +y 2 − x 2 − 2y
Page 86 and 87: R n R n • q = m
Page 88 and 89: ∂A ∂A
Page 90 and 91: Q = {0} × R y x = 0 Q0 = R
Page 92 and 93: D xy dx dy = = = = 1 dx √ 1−x
Page 94 and 95: x ′ xi i = 1−|x4| i = 1, 2, 3
Page 96 and 97: Area(E) = = = 1 2 = 1 4 E π 4
Page 98 and 99: (∆) = 2π dx dy dz = dθ ρ d
Page 100 and 101: Kn ⊂ Kn+1 n > 1 n Fp
Page 102 and 103: x ′2 + y ′2 = (ρ ′ (θ) cos
Page 104 and 105: (2, 0) A f dx dy = = = A A π =
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T R3 ·1 (
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xy z ∈ [0, 1] z C C(z) ≡
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(ρ, ϕɛ, θ) P = (x, y,
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Area (∂E2) = = = = = = ∂E2 π
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Φ + F (∂E1) Ψ 1 u
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F Φ + F (∂E) = = = Vol (E) =
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0 ≤ α < 2π 2π ω = γα =
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ω 1 R 2 \ {0} f(x, y) ω =
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• • • ∞ n=1 1 n 2 = ∞ n=1
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X ′′ (x) X(x) = −µ X(0) = X(
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{sin nx}n ∆v = 0 ⇐⇒ c ′
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• inf | sin z| = 0 4√ i( i = {
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2z + 3 z + 1 1 = 2 = ((z − 1) + 5
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1 = |z − a| 2 1 ρ 2 − |a|
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f R = {aω1 + bω2 : 0
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•
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c > 1 c = 1 Rez = 1 c < 1
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Resπ = −1 . Reskπ = (−1) k
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Res √ i 2 = i√2 Res √ i 3 =
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sin 2 z (0, π 2 ( π 2 ) ,
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R → ∞ z 2 − z + 2 CR
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π 2 i 2 ∞ = lim f(z) dz =
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f(0) = 0 g(z) = f(z) − f(0)
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1 n −n ∞ sin πz = π
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g = f ◦ S−1 f(z0
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un [a, b] un (a, b)
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f(x) = sin 2 (xy) x 2 +y 2 (x, y)
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2 y(x) (0, 1 2 ) {(x,
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Ac 1 + 3x2 dxdy + y2 Ac ≡ {(x, y
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ϕ : R ↦→ R 1 1 ϕ f(x)d
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P x P A, B, C, D ∈
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f : R 3 → R µf (x, r) = 1 4
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D = {(x, y, z) : x 2 + y 2 + z 2
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az + b T z = . T (R∪∞)
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cos z f(z) = z f(z) = e 1 z f(z)
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