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⎛<br />
det ⎝ n1<br />
n2<br />
n3<br />
∂xψ1<br />
∂xψ2<br />
∂xψ3<br />
∂yψ1<br />
∂yψ2<br />
∂yψ3<br />
⎛ −y<br />
⎞<br />
√ 1 0<br />
x2 +y2 +4<br />
⎞ ⎜<br />
⎟<br />
⎜<br />
⎟<br />
⎜ −x<br />
⎠ = det ⎜ √<br />
⎟<br />
0 1 ⎟<br />
⎜ x2 +y2 +4<br />
⎟<br />
⎜<br />
⎟<br />
⎝<br />
⎠<br />
√ 2 y/2 x/2<br />
x2 +y2 +4<br />
⎛<br />
⎞<br />
−y 1 0<br />
1<br />
= det ⎝ −x 0 1 ⎠ =<br />
x2 + y2 + 4 2 y/2 x/2<br />
2 + y2 /2 + x2 /2<br />
<br />
x2 + y2 + 4<br />
= 1<br />
x2 + y2 + 4 > 0.<br />
2<br />
<br />
F = (F1, F2, F3) <br />
Φ(S, ⎛<br />
<br />
F ) = det ⎝<br />
D<br />
F1<br />
⎞<br />
◦ ψ ∂xψ1 ∂yψ1<br />
<br />
F2 ◦ ψ ∂xψ2 ∂yψ2 ⎠ dxdy =<br />
⎛<br />
x<br />
det ⎝ y<br />
1<br />
0<br />
0<br />
1<br />
⎞<br />
⎠ dxdy<br />
F3 ◦ ψ ∂xψ3 ∂yψ3<br />
<br />
<br />
= (1 − xy) dxdy = Area(D) −<br />
D 1 y/2 x/2<br />
<br />
2π √<br />
2 3<br />
xy dxdy = 12π −<br />
r<br />
D<br />
D<br />
0 0<br />
2 <br />
cos θ sin θ ρ dρ dθ<br />
√<br />
2 3<br />
= 12π − r<br />
0<br />
3 2π<br />
dr · cos θ sin θ = 12π.<br />
12π<br />
0<br />
λ ∈ R λ < min{f(x, y) : x, y ∈ D} λ <br />
f D λ < 0 <br />
div F (x, y, z) = ∂F1<br />
∂x<br />
+ ∂F2<br />
∂y<br />
+ ∂F3<br />
∂z<br />
C S S − := D × {λ} <br />
C (0, 0, −1) C {(x, y, λ) : x, y, ∈ D} <br />
S<br />
<br />
= 2.<br />
L = {(x, y, z) : x 2 + y 2 = 12, λ ≤ z ≤ f(x, y)}.<br />
(x, y, 0)/ x2 + y2 <br />
<br />
<br />
div <br />
F dxdydz = F · ˆn dσ.<br />
<br />
C<br />
∂C<br />
2Volume(C) = Φ(S, <br />
F ) +<br />
<br />
2<br />
f(x,y)<br />
<br />
dz dxdy = Φ(S,<br />
D λ<br />
<br />
F ) −<br />
S −<br />
D<br />
<br />
F · ˆn dσ +<br />
<br />
dσ +<br />
L<br />
L<br />
F · ˆn dσ<br />
x 2 + y 2<br />
x 2 + y 2 dσ<br />
2Volume(C) = Φ(S, F ) − Area(D) + √ 12 Area(L)
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