I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce.

The author turns the left side of the equation into:

Just to be clear; is a vector valued function,

is a fixed vector,

are scalar valued functions.

The first part is fine:

The divergence of a fixed vector is zero and so,

Next I need to find

I am not sure what to do with is. is a scalar valued function, but so is I think . I know of the product rule between a scalar and a vector valued function, but what happens when there are two scalar valued functions?

Any suggestions are welcome, thanks.