 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ia/paper-3/4a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

\int_{V} \nabla \cdot F d V = \int_{\partial V} F \cdot n d S

for the outward normal. Here

\nabla \cdot (x e^{- r^{2}}) = (3 - 2 r^{2}) e^{- r^{2}}

, so radial integration gives

4 π a^{3} e^{- a^{2}}

. On the sphere

F \cdot n = a e^{- a^{2}}

, whose integral is the same

4 π a^{3} e^{- a^{2}}

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