Summarize Timeline Top Qs Fact Check On se place dans une variété pseudo-riemannienne M et on note
∇
{\displaystyle \nabla }
connexion de Levi-Civita . Pour un champ vectoriel
V
{\displaystyle \mathbf {V} }
∇
⋅
V
{\displaystyle \nabla _{\cdot }V}
; on appelle divergence de V la trace de ce champ, c'est un champ scalaire . Dans une carte quelconque, la valeur de ce champ est :
div
V
=
trace
(
∇
⋅
V
)
=
v
i
;
i
=
v
i
,
i
+
Γ
i
j
i
v
j
{\displaystyle {\text{div }}\mathbf {V} ={\text{trace}}(\nabla _{\cdot }V)=v^{i}{}_{;i}=v^{i}{}_{,i}+\Gamma _{ij}^{i}v^{j}}
Mettant à profit la formule de contraction
Γ
i
j
i
=
1
det
g
∂
j
det
g
{\displaystyle \Gamma _{ij}^{i}={\frac {1}{\sqrt {\det g}}}\partial _{j}{\sqrt {\det g}}}
on a
∇
v
=
1
det
g
∂
j
(
det
g
v
j
)
{\displaystyle \nabla \mathbf {v} ={\frac {1}{\sqrt {\det g}}}\partial _{j}\left({\sqrt {\det g}}\;v^{j}\right)}
Cette formule permet, une fois établi le tenseur métrique , de calculer facilement la divergence dans un système de coordonnées quelconque.
En coordonnées sphériques ,
la racine carrée du déterminant du tenseur métrique vaut
r
2
sin
θ
{\displaystyle r^{2}\sin \theta }
(
∇
⋅
v
)
i
=
1
r
2
sin
θ
∂
i
(
r
2
sin
θ
v
i
)
{\displaystyle (\nabla \cdot \mathbf {v} )^{i}={\frac {1}{r^{2}\sin \theta }}\partial _{i}\left(r^{2}\sin \theta \;v^{i}\right)}
Dans la base naturelle, on a
v
=
v
r
e
r
+
v
θ
e
θ
+
v
ϕ
e
ϕ
∇
⋅
v
=
(
2
r
+
∂
∂
r
)
v
r
+
(
1
tan
θ
+
∂
∂
θ
)
v
θ
+
∂
∂
ϕ
v
ϕ
{\displaystyle {\begin{matrix}\mathbf {v} &=&v^{r}\mathbf {e} _{r}&+&v^{\theta }\mathbf {e} _{\theta }&+&v^{\phi }\mathbf {e} _{\phi }\\\nabla \cdot \mathbf {v} &=&\left({\frac {2}{r}}+{\frac {\partial }{\partial r}}\right)v^{r}&+&\left({\frac {1}{\tan \theta }}+{\frac {\partial }{\partial \theta }}\right)v^{\theta }&+&{\frac {\partial }{\partial \phi }}v^{\phi }\end{matrix}}}
et donc dans la base orthonormée
(
e
r
,
e
θ
r
,
e
ϕ
r
sin
θ
)
{\displaystyle \left(\mathbf {e} _{r},{\frac {\mathbf {e} _{\theta }}{r}},{\frac {\mathbf {e} _{\phi }}{r\sin \theta }}\right)}
:
v
=
v
r
e
r
+
{
r
v
θ
}
{
e
θ
r
}
+
{
r
sin
θ
v
ϕ
}
{
e
ϕ
r
sin
θ
}
∇
⋅
v
=
(
2
r
+
∂
∂
r
)
v
r
+
(
1
r
tan
θ
+
1
r
∂
∂
θ
)
{
r
v
θ
}
+
1
r
sin
θ
∂
∂
ϕ
{
r
sin
θ
v
ϕ
}
{\displaystyle {\begin{matrix}\mathbf {v} &=&v^{r}\mathbf {e} _{r}&+&\left\{rv^{\theta }\right\}\left\{{\frac {\mathbf {e} _{\theta }}{r}}\right\}&+&\left\{r\sin \theta \;v^{\phi }\right\}\left\{{\frac {\mathbf {e} _{\phi }}{r\sin \theta }}\right\}\\\nabla \cdot \mathbf {v} &=&\left({\frac {2}{r}}+{\frac {\partial }{\partial r}}\right)v^{r}&+&\left({\frac {1}{r\tan \theta }}+{\frac {1}{r}}{\frac {\partial }{\partial \theta }}\right)\left\{rv^{\theta }\right\}&+&{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \phi }}\left\{{r\sin \theta \;v^{\phi }}\right\}\end{matrix}}}
En coordonnées cylindriques , la racine carrée du déterminant du tenseur métrique vaut
r
{\displaystyle r}
(
∇
⋅
v
)
i
=
1
r
∂
i
(
r
v
i
)
{\displaystyle (\nabla \cdot \mathbf {v} )^{i}={\frac {1}{r}}\partial _{i}\left(rv^{i}\right)}
Dans la base naturelle, on a
v
=
v
r
e
r
+
v
ϕ
e
ϕ
+
v
z
e
z
∇
⋅
v
=
(
1
r
+
∂
∂
r
)
v
r
+
∂
∂
ϕ
v
ϕ
+
∂
∂
z
v
z
{\displaystyle {\begin{matrix}\mathbf {v} &=&v^{r}\mathbf {e} _{r}&+&v^{\phi }\mathbf {e} _{\phi }&+&v^{z}\mathbf {e} _{z}\\\nabla \cdot \mathbf {v} &=&\left({\frac {1}{r}}+{\frac {\partial }{\partial r}}\right)v^{r}&+&{\frac {\partial }{\partial \phi }}v^{\phi }&+&{\frac {\partial }{\partial z}}v^{z}\end{matrix}}}
et donc dans la base orthonormée
(
e
r
,
e
ϕ
r
,
e
z
)
{\displaystyle \left(\mathbf {e} _{r},{\tfrac {\mathbf {e} _{\phi }}{r}},\mathbf {e} _{z}\right)}
:
v
=
v
r
e
r
+
{
r
v
ϕ
}
{
e
ϕ
r
}
+
v
z
e
z
∇
⋅
v
=
(
1
r
+
∂
∂
r
)
v
r
+
1
r
∂
∂
ϕ
{
r
v
ϕ
}
+
∂
∂
z
v
z
{\displaystyle {\begin{matrix}\mathbf {v} &=&v^{r}\mathbf {e} _{r}&+&\left\{rv^{\phi }\right\}\left\{{\frac {\mathbf {e} _{\phi }}{r}}\right\}&+&v^{z}\mathbf {e} _{z}\\\nabla \cdot \mathbf {v} &=&\left({\frac {1}{r}}+{\frac {\partial }{\partial r}}\right)v^{r}&+&{\frac {1}{r}}{\frac {\partial }{\partial \phi }}\left\{rv^{\phi }\right\}&+&{\frac {\partial }{\partial z}}v^{z}\end{matrix}}}
