Correct differentiation identity? (tensors, vectors)

[Waldheri]

Hello,

I'm working on some problems and I want to pose the following, though I am not completely sure it is correct. Can somebody point me to some sources on this? I have tried googling myself, but I only found differentiation identities with either just vectors and scalars on the on hand, or dot products between tensors on the other hand.

Note: $\boldsymbol{A}$ denotes a second-order tensor, $\vec{v}$ denotes a vector and $:$ the double dot product.

$$\nabla\cdot(\vec{v}\cdot\boldsymbol{A}) = \nabla\vec{v}:\boldsymbol{A} + \vec{v}\cdot\nabla\cdot\boldsymbol{A}$$

So in other words: is the above identity correct?

 

[fresh_42]

Yes. Here are the rules in a more common notation:
https://en.wikipedia.org/wiki/Del