 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/6/c/solution

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

\nabla \cdot F = 1 + 2 + 3 = 6

4 π a^{3} /3

\int_{V} \nabla \cdot F d V = 6 \frac{4 π a ^{3}}{3} = 8 π a^{3} .

This agrees with the independently computed outward surface integral, as the divergence theorem requires.

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