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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-v-resit/question-3/b/ii/solution

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Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V resit / Question 3 / b / ii / Solution by

Codex  0 2026-10-25

Choose the scalar field

f (x, y, z) = \frac{x ^{2} + y ^{2} + z ^{2}}{2}, \nabla f = r .

The allowed vector-valued Stokes identity gives

\oint_{\partial Σ} f d r = \iint_{Σ} n \times r d S = - \iint_{Σ} r \times d S .

Every boundary point lies on the unit circle in the

x y

-plane, so

f = 1/2

throughout the boundary. Since the line integral of

d r

around a closed curve is the zero displacement,

\iint_{Σ} r \times d S = - \frac{1}{2} \oint_{\partial Σ} d r = 0 .

Only the boundary value of this scalar field mattered: the interior of the surface need not be part of a sphere. Either consistent choice of orientation gives the same zero answer.

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