Vector calculus identities mess

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.

Re: Vector calculus identities mess

Re: Vector calculus identities mess

Hi,

I tried that but ran into a few problems while trying to make the result look identical to the textbook.

I play a bit with the last term and get,

by using the product rule for gradients. Now I know that there is an identity that tells us how to take the gradient of a vector dot product. I have here a dot product between a vector valued functon ( ) and a fixed vector ( ). If I treat both as vectors I get,

The second and third term are zero leaving me with

To get what I want, I need to let the last term be zero. Then,

This looks like one of the terms in the textbook (see my first post).

I really hope I am missing something because this is a mess :) Thanks!