What's the $\otimes$-operator in the proof of Reynolds' transport theorem at Wikipedia? In the proof of Reynolds' transport theorem at Wikipedia, they use the identities $$\nabla\cdot(v\otimes w)=v(\nabla\cdot w)+\nabla v\cdot w$$ and $$(a\otimes b)\cdot n=(b\cdot n)a\;,$$ where $n$ is probably the outer unit normal field of a given domain boundary. > But which operator is "$\otimes$" here? And why do these identities hold?

It is the outer product. If you regard $u$ and $v$ as column vectors it is defined as $u \otimes v = u v^T$.