You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
L −<br />
<br />
F · ˆn dσ = −<br />
L− √ 1 <br />
2<br />
√ dσ = −<br />
1 + 2z2 −1<br />
√ 1+2z2 0<br />
−2 √ 2<br />
ϕ = (ϕ1, ϕ2, ϕ3) <br />
Jac ϕ(θ, z) =<br />
⎛<br />
⎝ ∂θϕ1 ∂zϕ1<br />
∂θϕ2 ∂zϕ2<br />
∂θϕ3 ∂zϕ3<br />
⎛<br />
−<br />
⎞ ⎜<br />
⎠ ⎜<br />
= ⎜<br />
⎝<br />
√ 1 + 2z2 sin θ<br />
√<br />
1 + 2z2 cos θ<br />
√<br />
2<br />
√<br />
1 + 2z2 drdz = −√ 1<br />
2 dz = −2<br />
−1<br />
√ 2.<br />
2z cos θ<br />
√ 1 + 2z 2<br />
2z sin θ<br />
√ 1 + 2z 2<br />
0 1<br />
Σ Ω =] −<br />
π/4, π/4[×] − 1, 1[ <br />
Φ(Σ, <br />
F ) :=<br />
Σ<br />
<br />
F · ˆn dσ =<br />
Ω<br />
ω3( F ◦ ϕ, ∂1ϕ, ∂2ϕ, ∂3ϕ)(x) dx<br />
⎛<br />
F1 ◦ ϕ −<br />
⎜<br />
⎜<br />
= det ⎜<br />
Ω ⎜<br />
⎝<br />
√ 1 + 2z2 sin θ<br />
F2 ◦ ϕ √ 1 + 2z2 cos θ<br />
−1<br />
1<br />
(1 + 2z<br />
−π/4<br />
2 )z dθ<br />
π/4<br />
2z cos θ<br />
√ 1 + 2z 2<br />
2z sin θ<br />
√ 1 + 2z 2<br />
⎞<br />
⎟ (x) dx<br />
⎟<br />
⎠<br />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
<br />
F3 ◦ ϕ 0 1<br />
= F3 ◦ ϕ(θ, z) (−2z sin<br />
Ω<br />
2 θ − 2z cos 2 θ) dθdz+<br />
<br />
+<br />
<br />
F1 ◦ ϕ(θ, z)<br />
Ω<br />
1 + 2z2 cos θ + F2 ◦ ϕ(θ, z) 1 + 2z2 <br />
sin θ dθdz<br />
<br />
= −2 (ϕ<br />
Ω<br />
2 1(θ, z) + ϕ 2 2(θ, z))z dθdz+<br />
<br />
1 1 <br />
+ √ 1 + 2z2 cos θ + √ 1 + 2z2 sin θ dθdz<br />
Ω 1 + 2z2 1 + 2z2 <br />
= −2 (1 + 2z<br />
Ω<br />
2 <br />
)z dθdz + (cos θ + sin θ) dθdz<br />
1<br />
π/4<br />
Ω<br />
1<br />
π/4<br />
<br />
= −2<br />
dz +<br />
(cos θ + sin θ) dθ dz<br />
−1 −π/4<br />
= −π<br />
= 2<br />
−1<br />
π/4<br />
−π/4<br />
(1 + 2z 2 )z dz + 2<br />
cos θ dθ = 2 √ 2.<br />
−π/4<br />
(cos θ + sin θ) dθ
Change language