Curl and Product Rule

I know the famous identity

$\nabla \times (\vec{A} \times \vec{B}) = A(\nabla \cdot \vec{B}) -B(\nabla \cdot \vec{A}) + (B\cdot \nabla) \vec{A}- (A\cdot \nabla) \vec{B}$

My question: Can I distribute the curl like a normal differential operator as follows:

$\nabla \times (\vec{A} \times \vec{B}) =(\nabla \times \vec{A})\times B + (\nabla \times \vec{B})\times A$

With the del operator acting only on A in $(\nabla \times \vec{A})\times \vec{B}$ and only acting on B in the second term?

Or are there any "product rules" for the curl of a cross product other than the well-known one I gave in the beginning?

EDIT:

I posted the question because in the solution to my homework we are given the following equation which I don't understand:

$\nabla \times (\vec{e}_r \times \phi \vec{A}) = (\nabla \phi)\times (\vec{e}_r \times \vec{A})$ with the radial unit vector $\vec{e}_r$ and $\nabla \times \vec{A}=0$.

So does anyone have an idea where this comes from, then?

1 Answer

No. Choose some simple parallel vectors like $$\vec A = \vec B = \langle x, x, x\rangle$$ Then $\vec A \times \vec B = \vec 0$, but

$$\nabla \times \vec A = \langle 0, -1, 1\rangle$$ so that $$(\nabla \times \vec A) \times \vec B = \langle -2x, x, x\rangle$$

I'm not aware of any similar rules that involve only curl, although there is a product rule for multiplying by a scalar function, e.g. $\nabla (f \vec A)$. Some identities for second derivatives are given here as well.