I encountered the following claim in a book (Gockenbach: Understanding and Implementing the Finite Element Method), and I can't make sense of the equation.
"... the reader to verify that if $\sigma$ is a 2-tensor and $v$ is a vector, then $$ \nabla \cdot (\sigma v) = \left(\nabla \cdot \sigma^T \right) \cdot v + \sigma \cdot \nabla v^T $$
The issue that I am having with understanding this equation is that to me, the $ \nabla \cdot (\sigma v)$ term should be the divergence of a vector, or a scalar. The term $\left(\nabla \cdot \sigma^T \right) \cdot v$ to me seems like the dot product between 2 vectors, again a scalar. The last term $\sigma \cdot \nabla v^T$ seems to me to be the dot product between two order 2 tensors, again an order 2 tensor. Does anyone understand what the author means?
The dot between two tensors is being used to mean
$$\alpha\cdot\beta=\sum_{ij}\alpha_{ij}\beta_{ij}$$