The title says it. Is $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = 0$ if $\nabla^2 \chi = 0$?
$\chi$ is a field in $R^2$.
My attempt: I cannot get rid of this term by using any of the vector calculus identities, so I was thinking if there is some other reason as to why it might be zero. I do not know if the dot product is zero. It probably is not since at least as far as I can see there is no easy way to manipulate it into something that is trivially zero.
If it is not zero, I would still be happy if there is a vector $\mathbf{F}$ such that $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = \nabla \cdot \mathbf{F}$.