Which of the following vector differential identities is/are universally TRUE for twice continuously differentiable scalar fields $\phi$ and vector fields $\vec{F}$?
Options:
(A) $\nabla \times (\nabla \phi) = \vec{0}$ (curl of gradient is identically zero)
(B) $\nabla \cdot (\nabla \times \vec{F}) = 0$ (divergence of curl is identically zero)
All four are canonical identities of vector calculus:
1. $\nabla \times (\nabla \phi) = \vec{0}$ because mixed partials commute (A is true).
2. $\nabla \cdot (\nabla \times \vec{F}) = 0$ because divergence of any curl field is zero (B is true).
3. Vector Laplacian identity: $\nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - \nabla^2 \vec{F}$ (C is true).
4. Product rule for divergence: $\nabla \cdot (\phi \vec{F}) = \phi (\nabla \cdot \vec{F}) + \vec{F} \cdot (\nabla \phi)$ (D is true).