Vector Calculus

Which of the following vector calculus identities are identically TRUE for twice continuously differentiable scalar fields $\phi$ and vector fields $\vec{A}$?

Options:

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