 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/8/b/solution

Use the outward unit normal on the unit sphere and the convention that

θ

is measured from the positive vertical axis. Then

F \cdot n = sin θ

d S = sin θ d θ d ϕ

. The flux is therefore

\int_{S} F \cdot d S = \int_{0}^{2 π} \int_{0}^{π} sin^{2} θ d θ d ϕ = π^{2} .

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