For a solenoid vector field $F$, show that $\nabla\times(\nabla\times(\nabla\times(\nabla\times(F))))=\nabla^{4}F$

Question

For a solenoid vector field $F$, show that $\nabla\times(\nabla\times(\nabla\times(\nabla\times(F))))=\nabla^{4}F$

Now we have $\nabla \cdot \vec F=0$ : Therefore

$\nabla\times(\nabla\times \vec F)=\nabla(\nabla \cdot \vec F)-\nabla^2 \vec F\;$ but $\nabla(\nabla \cdot \vec F)=0$. Therefore

$\nabla\times(\nabla\times \vec F)=-\nabla^2 \vec F) \;$

Now calculating $\nabla\times(\nabla\times- \nabla^2 \vec F)=\nabla^4 \vec F) -\nabla(\nabla \cdot (\nabla^2 \vec F))$

I am not able to show that $\nabla(\nabla \cdot (\nabla^2 \vec F)))=0$

Answer 1

You can use the fact that $$\nabla \cdot (\nabla^2 F) = \nabla^2(\nabla \cdot F) \tag1 $$ to readily conclude. Equation $(1)$ can be shown by employing $$\nabla^2F= \nabla(\nabla \cdot F) - \nabla \times (\nabla \times F) $$ the identity $\nabla \cdot (\nabla \times F) \equiv 0$ and the definition $\nabla^2F := \nabla\cdot\nabla F $.