Example

Consider a sphere of radius $r$ centered at the origin of coordinates, and assume that the vector field $\underline{F}=\underline{r}$. Then $\nabla \cdot \underline{r}=3$ from example (A.2.3), hence the left-hand side (LHS) of (A.15) is simply three times the volume of the sphere, or $4 \pi r^3$. The integrand in the right-hand side is $\hat{n} \cdot \underline{F}=\hat{r} \cdot \underline{r}=r$, hence the integral equals $r$ times the surface of the sphere, or $4 \pi r^3$, QED.

Figure A.2: Geometry for the divergence theorem.