Calculus: Sequences, Infinite Series & Convergence

Which of the following differential operators and vector identities is/are MATHEMATICALLY VALID for smooth scalar field $\phi$ and vector field $\vec{F}$?

Options:

• (A) $\nabla \times (\nabla \phi) = \vec{0}$ (Curl of gradient is zero)
• (B) $\nabla \cdot (\nabla \times \vec{F}) = 0$ (Divergence of curl is zero)
• (C) $\nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - \nabla^2 \vec{F}$
• (D) $\nabla \cdot (\phi \vec{F}) = \phi (\nabla \cdot \vec{F}) + \vec{F} \cdot (\nabla \phi)$