On appelle flux (ou intégrale de surface ) du champ vectoriel
F
{\displaystyle \mathbf {F} }
R
3
{\displaystyle \mathbb {R} ^{3}}
orientée
Σ
{\displaystyle \Sigma }
Φ
≡
∫
Σ
F
⋅
d
S
{\displaystyle \Phi \equiv \int _{\Sigma }\mathbf {F} \cdot \mathrm {d} \mathbf {S} }
où
d
S
{\displaystyle \mathrm {d} \mathbf {S} }
vecteur normal élémentaire et
⋅
{\displaystyle \cdot }
produit scalaire . Si la surface est donnée par le paramétrage
σ
(
u
,
v
)
{\displaystyle \sigma (u,v)}
u
{\displaystyle u}
v
{\displaystyle v}
Ω
{\displaystyle \Omega }
d
S
=
[
∂
σ
∂
u
×
∂
σ
∂
v
]
d
u
d
v
{\displaystyle \mathrm {d} \mathbf {S} =\left[{\frac {\partial \sigma }{\partial u}}\times {\frac {\partial \sigma }{\partial v}}\right]\mathrm {d} u\,\mathrm {d} v}
et le flux est alors
Φ
=
∬
Ω
F
(
σ
(
u
,
v
)
)
⋅
[
∂
σ
∂
u
×
∂
σ
∂
v
]
d
u
d
v
=
∬
Ω
det
(
F
,
∂
σ
∂
u
,
∂
σ
∂
v
)
d
u
d
v
{\displaystyle \Phi =\iint _{\Omega }\mathbf {F} {\bigl (}\sigma (u,v){\bigr )}\cdot \left[{\frac {\partial \sigma }{\partial u}}\times {\frac {\partial \sigma }{\partial v}}\right]\mathrm {d} u\,\mathrm {d} v=\iint _{\Omega }\det \left(\mathbf {F} ,{\tfrac {\partial \sigma }{\partial u}},{\tfrac {\partial \sigma }{\partial v}}\right)\mathrm {d} u\,\mathrm {d} v}
Si
Σ
{\displaystyle \Sigma }
sans bord ) entourant un volume[ 1]
V
{\displaystyle V}
théorème de flux-divergence :
Φ
=
∮
Σ
F
⋅
d
S
=
∭
V
div
F
d
3
V
{\displaystyle \Phi =\oint _{\Sigma }\mathbf {F} \cdot {\rm {d}}\mathbf {S} =\iiint _{\mathcal {V}}\operatorname {div} \,\mathbf {F} \;{\rm {d}}^{3}V}
De la même manière, on définit le flux du champ
F
=
(
P
,
Q
)
{\displaystyle \mathbf {F} =(P,Q)}
R
2
{\displaystyle \mathbb {R} ^{2}}
Γ
{\displaystyle \Gamma }
Ψ
=
∫
Γ
F
⋅
d
n
=
∬
Γ
(
P
d
y
−
Q
d
x
)
{\displaystyle \Psi =\int _{\Gamma }\mathbf {F} \cdot \mathrm {d} \mathbf {n} =\iint _{\Gamma }(P\,\mathrm {d} y-Q\,\mathrm {d} x)}
où
d
n
=
(
d
y
,
−
d
x
)
{\displaystyle \mathrm {d} \mathbf {n} =(\mathrm {d} y,-\mathrm {d} x)}
F
{\displaystyle \mathbf {F} }
circulation (ou intégrale curviligne ) du champ orthogonal
G
=
(
−
Q
,
P
)
{\displaystyle \mathbf {G} =(-Q,P)}
Ψ
=
∫
Γ
G
⋅
d
r
{\displaystyle \Psi =\int _{\Gamma }\mathbf {G} \cdot \mathrm {d} \mathbf {r} }
avec
d
r
=
(
d
x
,
d
y
)
{\displaystyle \mathrm {d} \mathbf {r} =(\mathrm {d} x,\mathrm {d} y)}
