### Basic $\nabla$ relations and operations This page is for fast recap so one does not have to dive into the actual contents to do the calculations. $\nabla \cdot (\nabla f) = \nabla^{2} f$ $\nabla \cdot(\nabla \times \mathbf{F}) = 0 $ $\nabla \times (\nabla f) = 0$ $\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla\cdot \mathbf{F}) - \nabla^{2} \mathbf{F}$ $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot\mathbf{B})$ $\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} (\nabla \times \mathbf{B})$ $\nabla \cdot (\phi \mathbf{A}) = \phi (\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla \phi)$