 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-3/11a/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

For coordinates

x = x (q)

, the Jacobian is

J_{ij} = \partial x_{i} / \partial q_{j}

. Expanding after separating

x_{1} = r cos θ_{1}

from the remaining coordinates gives

∣ J_{n} ∣ = r sin^{n - 2} θ_{1} ∣ J_{n - 1} ∣.

Consequently

d V = r^{n - 1} \prod_{j = 1}^{n - 2} sin^{n - 1 - j} θ_{j} d r d θ_{1} \dots d θ_{n - 2} d ϕ,

and on

r = R

the surface element is obtained by omitting

d r

and replacing

r

R

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