Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 3 / ii

The vector calculus identity $\nabla \cdot (\mathbf{A}\times \mathbf{B})=\mathbf{B}\cdot (\nabla \times \mathbf{A})-\mathbf{A}\cdot (\nabla \times \mathbf{B})$ gives zero because a constant vector field has zero curl and $\nabla \times \mathbf{r}=0$. ThusEquivalently, the Cartesian components are $(a_{y}z-a_{z}y,a_{z}x-a_{x}z,a_{x}y-a_{y}x)$, and each component is independent of the coordinate used in its contribution to the divergence.