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dσ = det 2 B1 + det 2 B2 + det 2 B3 du dv<br />
<br />
Jac ϕ<br />
dσ = |∂uϕ(u, v) ∧ ∂vϕ(u, v)| du dv<br />
(x0, y0, z0) <br />
(u0, v0) ∈ I×J ϕ(u0, v0) = (x0, y0, z0)<br />
<br />
ˆn(x0, y0, z0) = ∂uϕ(u0, v0) ∧ ∂vϕ(u0, v0)<br />
|∂uϕ(u0, v0) ∧ ∂vϕ(u0, v0)|<br />
(u0, v0) <br />
v v/|v|<br />
Σ ϕ <br />
ˆn(x, y, z) <br />
(u, v) ∈ I × J ˆn ◦ ϕ(u, v) · ∂uϕ(u, v) ˆn ◦ ϕ(u, v) · ∂vϕ(u, v) <br />
<br />
<br />
det<br />
⎛<br />
⎝ n1 ◦ ϕ(u, v) ∂uϕ1(u, v) ∂vϕ1(u, v)<br />
n2 ◦ ϕ(u, v) ∂uϕ2(u, v) ∂vϕ2(u, v)<br />
n3 ◦ ϕ(u, v) ∂uϕ3(u, v) ∂vϕ3(u, v)<br />
<br />
<br />
(u, v) <br />
Σ (ū, ¯v) <br />
P (¯x, ¯y, ¯z) Σ (ū, ¯v) <br />
ϕ(ū, ¯v) = P (¯x, ¯y, ¯x) (ū, ¯v)<br />
Σ f(x, y, z) = 0 ∇f(x, y, z)<br />
Σ Σ <br />
Σ ∇f <br />
Σ <br />
ϕ f = 0 <br />
∇f <br />
∂uϕ ∧ ∂vϕ<br />
F <br />
<br />
⎛<br />
<br />
<br />
F · ˆn dσ = det ⎝ F1<br />
⎞<br />
◦ ϕ ∂uϕ1 ∂vϕ1<br />
F2 ◦ ϕ ∂uϕ2 ∂vϕ2 ⎠ du dv<br />
S<br />
I<br />
J<br />
F3 ◦ ϕ ∂uϕ3 ∂vϕ3<br />
γ G γ : [0, T ] → R 3 C 1<br />
<br />
γ<br />
G · ds =<br />
T<br />
0<br />
G(γ(t)) · ˙γ(t) dt.<br />
Σ ψ : [a, b]×[c, d] → R 3 <br />
γ1(t) = ψ(t, c) t ∈ [a, b]<br />
γ2(t) = ψ(b, t) t ∈ [c, d] γ3(t) = ψ(b + a − t, d) t ∈ [a, b] γ4(t) = ψ(a, d + c − t) <br />
t ∈ [a, b] ψ [a, b] × [c, d]<br />
<br />
ψ(a, t) = ψ(b, t) t ∈]c, d[ ψ(t, c) = ψ(t, d) t ∈]a, b[ <br />
<br />
⎞<br />
⎠ .