Use suffix notation in Cartesian coordinates to establish the following two identities for the vector field $\mathbf{v}$ :

$$\nabla \cdot(\nabla \times \mathbf{v})=0, \quad(\mathbf{v} \cdot \nabla) \mathbf{v}=\nabla\left(\frac{1}{2}|\mathbf{v}|^{2}\right)-\mathbf{v} \times(\nabla \times \mathbf{v})$$