Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 4B / Solution

Differentiation of $x=\rho \cos \varphi$, $y=\rho \sin \varphi$ givesThe two radial face areas are ${\rho }_0\Delta \varphi \Delta z$ and $({\rho }_0+\Delta \rho )\Delta \varphi \Delta z$. Each constant-$\varphi$ face has area $\Delta \rho \Delta z$, and each constant-$z$ face has areaThe volume is this last expression times $\Delta z$.

Only the radial faces contribute to the flux of $F(\rho )e_{\rho }$. Dividing their difference by the volume and taking $\Delta \rho \to 0$ givesat $\rho ={\rho }_0$.