Vector Calculus

Which of the following vector identities are mathematically VALID for any sufficiently smooth scalar field $\phi$ and vector field $\vec{A}$?

Options:

• (A) $\nabla \times (\nabla \phi) = \vec{0}$ (curl of gradient is always zero)
• (B) $\nabla \cdot (\nabla \times \vec{A}) = 0$ (divergence of curl is always zero)
• (C) $\nabla \cdot (\nabla \phi) = \nabla^2 \phi$ (Laplacian of scalar field)
• (D) $\nabla \times (\nabla \times \vec{A}) = \vec{0}$ for all vector fields