 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-2/4/b/solution

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

\nabla \cdot F = 2 x yz + 2 x yz + 2 x yz = 6 x yz

. With outward orientation, the divergence theorem converts the surface flux into a volume integral:

\int_{S} F \cdot d S = 6 \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} x yz d x d y d z = 6 (\frac{1}{2})^{3} = \frac{3}{4} .

Directly, the three faces lying in coordinate planes contribute zero, and each opposite face contributes

1/4

, confirming the sign and normalization.

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