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

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I − T f ′ (y)∞,∞ = max {|2y1 cos y2| + |y 2 1 sin y2| , |2y1|} ≤ 3ρ ρ = 1 6 r = ρ 1 2T = 12 f(x0, g(x0)) = 0 ⇐⇒ g(x0) 2 − 6g(x0) − 16 = 0 ⇐⇒ g(x0) = −2 g(x0) = 8 P+ = (1, −2, 8) e P− = (1, −2, −2). ∂ f ∂ y (P+) = (2y − 6) |y=8 = 10 = 0 ∂ f ∂ y (P−) = (2y − 6) |y=−2 = −10 = 0 g± C∞ x0 f(x, g±(x)) = 0 g+(x0) = 8 e g−(x0) = −2 . x0 g+ x0 ∇g(x0) = 0 x0 f(x, g+(x)) = |x| 2 + g(x) 2 − 2x1 + 4x2 − 6g(x) − 11 ≡ 0 , 0 ≡ ∇(f(x0, g+(x0))) = = 2(x0) T −2 + 2g+(x0) ∇g+(x0) + − 6∇g+(x0) = 4 1 −2 = 2 + 16 ∇g+(x0) + − 6∇g+(x0) = −2 4 = 10 ∇g+(x0)
∇g+(x0) = 0 x0 Hg+ (x0) 0 ≡ ∂2 f(x, g(x)) = ∂ x2 2 0 = + (2g+(x0) − 6)Hg+ 0 2 (x0) + ∂x1g+(x0)∂x1g+(x0) ∂x1g+(x0)∂x2g+(x0) + 2 = ∂x2g+(x0)∂x1g+(x0) ∂x2g+(x0)∂x2g+(x0) 2 0 = + 10 Hg+(x0) , 0 2 Hg+ (x0) = ⎛ ⎜ ⎝ − 1 0 5 0 − 1 5 f f ∈ C ∞ (R n , R n ) ∂f ∂x (x) = In + 2(cos |x| 2 )A Aij ≡ (vixj) ∂f ∂x (0) = In , det ∂f ∂x (0) = det In = 1 . g f g ∈ C ∞ (Br(0)) r > 0 r ρ sup In ∂f − In ∂x |x|≤ρ ≤ 1 2 ∂fi = δi,j + 2vixj cos |x| ∂xj 2 δi,j 1 se i = j δi,j = 0 se i = j ⎞ ⎟ ⎠
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Rn Rn Rn
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Rn Rn Rn
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ℓ 1 ℓ ∞ x = {xn}n ∈ R N
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P ≡ {(x, y) ∈ R 2 y = x 2 (x
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∂ 5 f (1, 1, .., 1) n > 5 ∂x1
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z = z(x, y) (1, 1) z 3 −
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K ⊂ R2 \ y = 1 2 0 < δ < β
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(i) ∞ n=0 n 3 2 n (ii) ∞ n=1 n
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v ∈ R n 1 f(x) ≡ x +
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X ⊂ R n o X = ∅ X Qx ,
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E = {(x, y) ∈ R 2 : x ≥ 0, (x 2
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R 3 Γ ≡ {(u(t), v(t)) : t ∈ (
Page 26 and 27:
F F (x, y, z) = (x, y, z) ;
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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 56 and 57: F ′ (t) = ∂ f ∂ x (g(t), 1
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 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 =
Page 106 and 107: T R3 ·1 (
Page 108 and 109: xy z ∈ [0, 1] z C C(z) ≡
Page 110 and 111: (ρ, ϕɛ, θ) P = (x, y,
Page 112 and 113: Area (∂E2) = = = = = = ∂E2 π
Page 114 and 115: Φ + F (∂E1) Ψ 1 u
Page 116 and 117: F Φ + F (∂E) = = = Vol (E) =
Page 118 and 119: 0 ≤ α < 2π 2π ω = γα =
Page 120 and 121: ω 1 R 2 \ {0} f(x, y) ω =
Page 122 and 123: • • • ∞ n=1 1 n 2 = ∞ n=1
Page 124 and 125: X ′′ (x) X(x) = −µ X(0) = X(
Page 126 and 127: {sin nx}n ∆v = 0 ⇐⇒ c ′
Page 128 and 129: • 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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