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

In Cartesian components, $\mathbf{a}\cdot \mathbf{r}=a_{x}x+a_{y}y+a_{z}z$, with constant $a_i$. Taking the gradient component by component givesThe constancy of $\mathbf{a}$ matters: if it depended on position, derivatives of its components would also contribute.

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.

Away from the origin, the product rule and $\nabla r=\mathbf{r}/r$ giveThe field is undefined at $r=0$. It has outward flux $4\pi$ through any sphere centred there, so its global distributional divergence is $4\pi {\delta }^{(3)}(\mathbf{r})$, not the zero distribution. The pointwise result must not be extended through the singularity when applying the divergence theorem.