En coordonnées cylindriques, la racine carrée du déterminant du tenseur métrique vaut r et la divergence d'un champ de vecteurs s'écrit ∇⋅v=1r∂i(rvi){\displaystyle \nabla \cdot \mathbf {v} ={\frac {1}{r}}\partial _{i}\left(rv^{i}\right)}.
Dans la base naturelle, on a
v=vrer+vϕeϕ+vzez∇⋅v=(1r+∂∂r)vr+∂∂ϕvϕ+∂∂zvz{\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 (er,eϕr,ez){\displaystyle \left(\mathbf {e} _{r},{\frac {\mathbf {e} _{\phi }}{r}},\mathbf {e} _{z}\right)}:
v=vrer+{rvϕ}{eϕr}+vzez∇⋅v=(1r+∂∂r)vr+1r∂∂ϕ{rvϕ}+∂∂zvz{\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}}}