Which of the following statements are sensible and true about arbitrary vector field $\vec{F}$ and arbitrary multivariate function $f(x,y,z)$?

(I) $\nabla \cdot (\nabla \cdot \vec{F}) = 0$

(II) $\nabla \times (\nabla \cdot \vec{F}) = 0$

(III) $\nabla \cdot ( \nabla f ) = 0$

(IV) $\nabla \times ( \nabla f ) = 0$

(V) $\nabla \cdot ( \nabla \times \vec{F} ) = 0$

(VI) $\nabla \times ( \nabla \times \vec{F} ) = 0$

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