Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 11A / Solution

For coordinates $x=x(q)$, the Jacobian is $J_{ij}=\partial x_i/\partial q_j$. Expanding after separating $x_1=r\cos {\theta }_1$ from the remaining coordinates givesConsequentlyand on $r=R$ the surface element is obtained by omitting $dr$ and replacing $r$ by $R$.