Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 6

Take the outward orientation of a surface. The divergence is $\nabla \cdot \mathbf{F}=y^2+z^2+x^2$, so the divergence theorem converts the closed surface flux into an integral over the box:Each term separates; for example the $x^2$ term is $(2a^3/3)(2b)(2c)$. Summing givesHere $a,b,c>0$ are the box half-lengths. Reversing orientation of a surface would negate the result.

Differentiate the ellipsoidal parametrization. Its outward vector surface element isFor $a,b,c>0$, its dot product with $\mathbf{r}$ simplifies to $abc\sin \theta$. Thus the requested direct surface integral isAs a check, the divergence of $\mathbf{r}$ is three, and the ellipsoid has volume $4\pi abc/3$, so the divergence theorem gives the same value.