 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-1/6/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Differentiate the ellipsoidal parametrization. Its outward vector surface element is

r_{θ} \times r_{ϕ} = (b c sin^{2} θ cos ϕ, a c sin^{2} θ sin ϕ, ab sin θ cos θ) .

For

a, b, c > 0

, its dot product with

r

simplifies to

ab c sin θ

. Thus the requested direct surface integral is

\int_{S} r \cdot d S = ab c \int_{0}^{2 π} \int_{0}^{π} sin θ d θ d ϕ = 4 πab c .

As a check, the divergence of

r

is three, and the ellipsoid has volume

4 πab c /3

, so the divergence theorem gives the same value.

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