By using suffix notation, prove the following identities for the vector fields $\mathbf{A}$ and B in $\mathbb{R}^{3}$ :

$$\begin{gathered} \nabla \cdot(\mathbf{A} \times \mathbf{B})=\mathbf{B} \cdot(\nabla \times \mathbf{A})-\mathbf{A} \cdot(\nabla \times \mathbf{B}) \\ \nabla \times(\mathbf{A} \times \mathbf{B})=(\mathbf{B} \cdot \nabla) \mathbf{A}-\mathbf{B}(\nabla \cdot \mathbf{A})-(\mathbf{A} \cdot \nabla) \mathbf{B}+\mathbf{A}(\nabla \cdot \mathbf{B}) \end{gathered}$$