 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ia/paper-3/12c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

The curl is constant:

\nabla \times F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x}) = (1, 1, 1) .

Orient the spanning planar disk by the unit normal

n = (a, b, c)

and orient its boundary by the right-hand rule, equivalently counterclockwise when viewed from the side towards which

n

points. Stokes theorem converts the line integral to a constant flux integral through a disk of area

π R^{2}

. With this orientation

\oint_{C} F \cdot d x = π R^{2} (a + b + c) .

Reversing the orientation changes the sign. The centre of the circle does not enter because the curl is constant.

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