Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 1 / Solution

For a continuously differentiable vector field on a bounded region with suitably piecewise smooth boundary, the divergence theorem giveswhere the normal vector points outward. In three dimensions, the position vector is $\mathbf{r}=(x,y,z)$ and $\nabla \cdot \mathbf{r}=1+1+1=3$. ThereforeThe result depends only on the enclosed volume, not on the shape or its location relative to the origin. Reversing the surface orientation reverses the sign.