En coordonnées sphériques, la racine carrée du déterminant du tenseur métrique vaut r sin θ {\displaystyle r\;\sin \theta } et la divergence d'un champ de vecteurs s'écrit ( ∇ ⋅ 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}}}
et la
divergence d'un champ de vecteurs s'écrit
.
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