# External Algebra – A Vector Computation Formula

I have unsuccessfully asked this question about mathematics. Let $$A, B$$ to be vector fields in $$mathbb R ^ 3$$. We have
$$text {curl} bigl ((A cdot nabla) B bigr) = (A cdot nabla) text {curl} B – (( text {curl} A) cdot nabla) B + R (A, B).$$
I know it $$R (A, A) = 0$$ and I guess $$R$$ should look like
$$d a_j wedge d b_j = text {curl} A times text {curl} B + text {div} A text {curl} B – text {div} B text {curl} A.$$
Is there a simple closed form for $$R (A, B)$$?