Vector Calculus

Which of the following vector identities are TRUE for smooth scalar field $\phi$ and vector field $\vec{F}$?

Options:

• (A) $\nabla \times (\nabla \phi) = 0$
• (B) $\nabla \cdot (\nabla \times \vec{F}) = 0$
• (C) $\nabla \cdot (\nabla \phi) = \nabla^2 \phi$
• (D) $\nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - \nabla^2 \vec{F}$