I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$ where $\overline{T}$ is a second order symmetric tensor and $\times$ is the vectorial product.
Once I integrated by part I want to use the divergence theorem to obtain a surface integral. It works very well in cartesian coordinates but I cannot manage to do it properly in cylindrical.
In cylindrical coordinates $\hat{\boldsymbol{\rho}},\hat{\boldsymbol{\theta}},\hat{\mathbf{z}}$,
\begin{align}&\nabla \cdot \overline{T}= \left[ \frac{1}{\rho} \frac{\partial}{\partial \rho}(\rho T_{\rho\rho}) + \frac{1}{\rho} \frac{\partial T_{\theta \rho}}{\partial \theta} + \frac{\partial T_{z \rho }}{\partial z} - \frac{T_{\theta \theta}}{\rho} \right] \hat{\boldsymbol{\rho}} + \\ &\qquad\qquad\left[ \frac{1}{\rho} \frac{\partial}{\partial \rho}(\rho T_{\rho\theta}) + \frac{1}{\rho} \frac{\partial T_{\theta\theta}}{\partial \theta} + \frac{\partial T_{z\theta}}{\partial z} + \frac{T_{\theta \rho }}{\rho} \right] \hat{\boldsymbol{\theta}} + \\ &\qquad\qquad\left[ \frac{1}{\rho} \frac{\partial}{\partial \rho}(\rho T_{\rho z}) + \frac{1}{\rho} \frac{\partial T_{\theta z}}{\partial \theta} + \frac{\partial T_{z z}}{\partial z} \right] \hat{\mathbf{z}} \end{align}
I tried a lot of different ways (5 days I am trying to do it) of doing the integration by part, but by checking numerically I know that my results is wrong. The value I obtain numerically seems to be more like ($\overline{T}$ is symmetric) \begin{align} \int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r}&= \oint \vec{r} \times \left( \hat{\boldsymbol{\rho}} \cdot \overline{T} \right) ~dS + \\ &\left( \int T_{\rho z} ~d^3\vec{r} - \int_0^{\rho_\text{max}} \left[ T_{\theta z}(2\pi) - T_{\theta z}(0) \right] ~\rho d\rho \right) \hat{\boldsymbol{\theta}} + \\ &\int_0^{\rho_\text{max}} \left[ T_{\theta\theta}(2 \pi) - T_{\theta\theta}(0) \right] \rho d\rho \hat{\mathbf{z}} \end{align} First, I am not sure of the result and second, I would like to know how to derive this analytically. Someone can help me? thank you very much