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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/19/c/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 19 / c / Solution by

Codex  0 2026-10-18

Take the sphere's outward normal. Since

F = k (x, y, z)

with constant

k

, its divergence is

3 k

. The divergence theorem gives

\int_{S} F \cdot d S = 3 k \frac{4 π r ^{3}}{3} = 4 πk r^{3} .

(1)

The translated centre does not enter the answer, because the divergence is constant. Equivalently the constant field produced by translating coordinates has zero net flux through a closed sphere. The inward orientation would negate the result.

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