I am working on deriving the Navier-Stokes equation in spherical coordinates for a homework assignment, but I've hit a serious math roadblock. My background in tensors is very minimal and a crucial term in the Navier-Stokes equation involves the divergence of a tensor product, \begin{equation*} \nabla \cdot \big(\rho \vec{v}\,\otimes\,\vec{v}\big). \end{equation*}
I saw on Wikipedia that, \begin{equation*} \nabla\cdot\big(\vec{B}\,\otimes\,\hat{A}\big) = \hat{A}\big(\nabla\cdot\vec{B}\big) + \big(\vec{B}\cdot\nabla\big)\hat{A}. \end{equation*} I am honestly not sure what $\hat{A}$ means, but I assume it's just another notation for $\vec{A}$. If so, this places me in a predicament because I now have to compute the gradient of a vector, $\nabla\vec{A}$, which I don't know how to do and cannot find any online resources describing how to do this in a way I can understand.
It would help me tremendously to have some kind of example of a tensor product and/or divergence of a tensor product that uses simple cartesian coordinates $x$, $y$, and $z$. I am confident that I can produce the equivalent in spherical coordinates, but as of now I am mathematically (not physically) stumped. Please help!