Vector calculus identities

Statement

Let $U \subseteq \mathbb{R}^3$ be open, let $\phi, \psi \colon U \to \mathbb{R}$ be scalar fields, and let $A \colon U \to \mathbb{R}^3$ be a vector field. If these fields are of class $C^1$, then $$\nabla (\phi \psi) = \phi \nabla \psi + \psi \nabla \phi, \qquad \nabla \cdot (\psi A) = \psi \, \nabla \cdot A + \nabla \psi \cdot A, \qquad \nabla \times (\psi A) = \psi \, \nabla \times A + \nabla \psi \times A.$$ If these fields are of class $C^2$, then $$\nabla \times (\nabla \phi) = 0, \qquad \nabla \cdot (\nabla \times A) = 0, \qquad \nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \Delta A,$$ where $\Delta A$ applies the Laplacian to each component of $A$.