 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/11/b/ii/solution

The scalar integrand still equals

10 e

throughout the unit sphere, and the upper hemisphere has the same surface area

2 π

. Excluding its equator changes no area. Therefore

\int_{S^{'}} (\nabla \cdot v) d S = 20 π e .

There is no sign reversal between hemispheres because this scalar surface integral has no choice of normal orientation.

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