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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/13/e/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 13 / e / Solution by

Codex  0 2026-10-18

The upward unit vector extracts the vertical curl component:

(\nabla \times F) \cdot k = 2 a .

(1)

It is constant over the unit disc, so

I = \int_{D} 2 a d S = 2 πa .

(2)

This agrees with part (d) by Stokes theorem: the boundary circle has counterclockwise orientation when viewed from the upward normal, including for the parametrizations

2 π t^{n}

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