 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021-resit/8/c/solution

Away from the origin, write

F = r^{- β - 1} r

\nabla \times F = \nabla (r^{- β - 1}) \times r + r^{- β - 1} \nabla \times r = - (β + 1) r^{- β - 3} r \times r = 0.

Consequently the surface integral of curl over the upper hemisphere is zero. Its outward unit normal vector induces a counterclockwise equator when viewed from above. Parametrize that boundary by

r (ϕ) = r_{2} (cos ϕ, sin ϕ, 0), d r = r_{2} (- sin ϕ, cos ϕ, 0) d ϕ .

\oint_{\partial S} F \cdot d r = 0 = \iint_{S} (\nabla \times F) \cdot \hat{n} d S .

This verifies Stokes theorem directly without passing through the singular origin.

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