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

∂S = ∂S1 ∪
∂S2
∂S1 = {(cos t, sin t, 0), 0 ≤ t < 2π}
∂S2 = {(cos t, sin t, 1), 0 ≤ t < 2π} .
∂S
∂S1
∂S2
+∂S
ω =
=
=
+∂S1
ω +
ω −
−∂S2
+∂S1 +∂S2
2 2
cos t + sin t
2π
0
= π.
1
ω =
ω =
− cos2 t + sin 2
t
dt =
2
F (x, y, z) = (F1, F2, F3) =
−y
x2 + y2 ,
+ z2 F3 = 0
F = (−∂zF2, ∂zF1, ∂xF2 − ∂yF1).
S
x
x2 + y2 , 0
+ z2
.
S = {Φ(t, z) = (cos t, sin t, z) : t ∈ [0, 2π), z ∈ [0, 1]} ;
ν = (cos t, sin t, 0)
∂tΦ ∧ ∂zΦ = 1
∂S
ω =
=
F × ν dσ =
S
2π 1
dt
0
= π .
0
2z
(1 + z2 =
) 2
C
C = A ∪ B A ≡ D × [1, 4]
z = 1, z = 4 D R 2 B
{(x, y, z) ∈ R 3 : (x, y) ∈ D, x 2 + y 2 ≤ z ≤ 1}