Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 11B (Vector Calculus)

SetThe ellipsoid becomes the unit ball and the Jacobian determinant is $abc$. Thereforeassuming $a,b,c>0$.

Fix the first index $i$ and form the vector field with components $F_j=T_{ij}$. The divergence theorem givesSubstitution yields the tensor divergence theorem

For $u=(-y,x,\alpha )$, one has $\nabla \cdot u=0$, and therefore

Close the upper half-ellipsoid with its unit-disk base $D$ in the plane $z=0$. By symmetry,On the base, the outward normal is $n=(0,0,-1)$, soThe first two components integrate to zero over the disk, while the third integrates to $-\pi {\alpha }^2$. The closed-surface tensor identity therefore givesMultiplying by ${\beta }^2$,