Annexesmathématiques
284
Gradient
∇ =
∂
∂
+
∂
∂
+
∂
∂
f
f
r
e
r
f e
r
f e
r
1
1
θ
θ ϕ
θ
ϕ
sin
∇ =
∂
∂
∂
∂
−
∂
∂
−
∂
∂
∂
∂
u
u
r r
u
u
r
r
u
u
r
u
r r
u
r
r
r

1
1
1
θ
θ ϕ
θ
θ
ϕ
θ
θ
sin
+ +
∂
∂
−
∂
∂
∂
∂
∂
∂
u
r
r
u
u
r
u
r
r
u
r
u
r
1
1
1
sin
sin
θ ϕ
θ
θ
θ
θ
ϕ
ϕ
ϕ
ϕ
cotg
ϕ ϕ
θ
θ
+
+




















u
r
u
r
r
  cotg
Divergence
∇ ⋅ =
∂ (
)
∂
+
∂ (
)
∂
+
∂
u r
r u
r
r
u
r
u
r
1
1
1
2
2
sin
sin
sin
θ
θ
θ
θ
θ
ϕ
∂ ∂ϕ
Laplacien
Δf r r
r
f
r
r
f
r
=
∂
∂
∂
∂
( ) +
∂
∂
∂
∂
( ) +
∂
1
1
1
2
2
2
2
sin
sin
sin
θ θ
θ θ
θ
2 2
2
f
∂ϕ
Rotationnel
∇ ∧ =
∂ (
)
∂
−
∂ ( )
∂






u
r
r
u
r u e r
1
1
2 sin
sin
θ
θ
θ
ϕ
ϕ
θ
r r
u
r
u
r
e
r
r u
r
r
sin
sin
θ ϕ
θ ϕ
θ
θ
∂
∂
−
∂ (
)
∂






∂ ( )
∂
−
∂
1
u u e
r
∂


























θ
ϕ
dérivée ParTiculaire
Coordonnées cartésiennes
d
d
f
t
f
t
u
f
x
u
f
y
u
f
z
x
y
z
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
Coordonnées cylindriques
d
d
f
t
f
t
u
f
r
u
r
f u
f
z
r
z
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂


Coordonnées sphériques
d
d
f
t
f
t
u
f
r
u
r
f
u
r
f
r
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
θ
ϕ
θ
θ ϕ
sin
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