Laplacian of cross product of two vectors

Question

Suppose A and B are two vectors then what is

$\nabla^2 (\mathbf{A} \times \mathbf{B}$)

I tried finding it on the internet but no luck.

Answer 1

Note that the cross-product satisfies Leibniz's formula in the form $$ \frac{\partial}{\partial x} ({\bf A} \times {\bf B}) = \frac{\partial {\bf A}}{\partial x} \times {\bf B} + {\bf A} \times \frac{\partial {\bf B}}{\partial x}$$ and thus $$ \frac{\partial^2}{\partial x^2} ({\bf A} \times {\bf B}) = \frac{\partial^2 {\bf A}}{\partial x^2} \times {\bf B} + 2 \frac{\partial {\bf A}}{\partial x} \times \frac{\partial {\bf B}}{\partial x} + {\bf A} \times \frac{\partial^2 {\bf B}}{\partial x^2}$$ Similarly for $y$ and $z$. Add them to get $\nabla^2 ({\bf A} \times {\bf B})$.