Related to the question Compute $a\otimes a$ in cylindrical coordinates consider \begin{align} A(\rho,\varphi,z)=\begin{pmatrix} a_{11}(\rho,\varphi,z) & a_{12}(\rho,\varphi,z) & a_{13}(\rho,\varphi,z)\\\\ a_{21}(\rho,\varphi,z) & a_{22}(\rho,\varphi,z) & a_{23}(\rho,\varphi,z)\\\\ a_{31}(\rho,\varphi,z) & a_{32}(\rho,\varphi,z) & a_{33}(\rho,\varphi,z). \end{pmatrix} \end{align}
Task Compute the divergence $div_{\text{cartesian}}$ in cartesian coordinates: We can compute \begin{align} \nabla_{\text{cartesian}} ~~\rho =\begin{pmatrix}\cos(\varphi)\\\\ \sin(\varphi)\\\\ 0\end{pmatrix}, \nabla_{\text{cartesian}} ~~\varphi =\frac{1}{\rho}\begin{pmatrix}-\sin(\varphi)\\\\ \cos(\varphi)\\\\ 0\end{pmatrix}, \nabla_{\text{cartesian}} ~~z =\begin{pmatrix}0\\\\ 0\\\\ 1\end{pmatrix} \end{align} and \begin{align} div_{\text{cartesian}}(A)&=\begin{pmatrix}\partial_x a_{11}(\rho,\varphi,z)+\partial_y a_{12}(\rho,\varphi,z)+\partial_z a_{13}(\rho,\varphi,z)\\\\ a_{21}(\rho,\varphi,z)+\partial_y a_{22}(\rho,\varphi,z)+\partial_z a_{23}(\rho,\varphi,z)\\\\ a_{31}(\rho,\varphi,z)+\partial_y a_{32}(\rho,\varphi,z)+\partial_z a_{33}(\rho,\varphi,z) \end{pmatrix} \end{align}
The divergence is taken rowwise. So for the first row we get \begin{align} \partial_x a_{11}(\rho,\varphi,z)+\partial_y a_{12}(\rho,\varphi,z)+\partial_z a_{13}(\rho,\varphi,z)&=\partial_{\rho}a_{11}\partial_{x}\rho+\partial_{\varphi}a_{11}\partial_{x}\varphi+\partial_{z}a_{11}\partial_{x}z\newline &+\partial_{\rho}a_{12}\partial_{x}\rho+\partial_{\varphi}a_{12}\partial_{x}\varphi+\partial_{z}a_{12}\partial_{x}z\newline &+\partial_{\rho}a_{13}\partial_{x}\rho+\partial_{\varphi}a_{13}\partial_{x}\varphi+\partial_{z}a_{13}\partial_{x}z\newline &=\partial_r a_{11}\cos(\varphi)-\partial_{\varphi}a_{11}\frac{\sin(\varphi)}{r}\newline &+\partial_r a_{12}\sin(\varphi)+\partial_{\varphi}a_{11}\frac{\cos(\varphi)}{r}\newline &+\partial_z a_{13} \end{align}
Doing the same for the second and third row \begin{align} div_{\text{cartesian}}(A)=\begin{pmatrix}\partial_r a_{11}\cos(\varphi)-\partial_{\varphi}a_{11}\frac{\sin(\varphi)}{r} +\partial_r a_{12}\sin(\varphi)+\partial_{\varphi}a_{12}\frac{\cos(\varphi)}{r}+\partial_z a_{13}\\\\ \partial_r a_{21}\cos(\varphi)-\partial_{\varphi}a_{21}\frac{\sin(\varphi)}{r} +\partial_r a_{22}\sin(\varphi)+\partial_{\varphi}a_{22}\frac{\cos(\varphi)}{r}+\partial_z a_{23}\\\\ \partial_r a_{31}\cos(\varphi)-\partial_{\varphi}a_{31}\frac{\sin(\varphi)}{r} +\partial_r a_{32}\sin(\varphi)+\partial_{\varphi}a_{32}\frac{\cos(\varphi)}{r}+\partial_z a_{33} \end{pmatrix} \end{align}
Question Is this correct?