The book I am reading gives a list of product rules, among them the following:
$$\nabla \cdot (v\times w) =(\nabla \cdot v) w-v\nabla \cdot w.$$
However, the left-hand side is a number whereas the right-hand side is a vector.
So clearly, something is wrong, maybe a dot should be replaced by a cross product, but I have not found a similar correct expression.
What is the formula that the authors tried to write?
The only product rule that I know for this situation is $$ \nabla\cdot(v\times w)=w\cdot(\nabla\times v)-v\cdot(\nabla\times w). $$