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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-v/1/b/solution

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Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 1 / b / Solution by

Codex  0 2026-10-25

Green's theorem for a positively oriented simple closed planar curve states

\oint_{C} (P d x + Q d y) = \iint_{D} (Q_{x} - P_{y}) d x d y .

(1)

Here

Q_{x} - P_{y} = 1 - (- 1) = 2

. The trapezium has parallel sides of lengths two and one, separated by height one, so its area is

3/2

. Therefore

\int_{C} F \cdot d r = 2 \cdot \frac{3}{2} = 3.

(2)

Every congruent trapezium gives the same answer when its boundary is traversed anticlockwise, since the scalar curl is constant and geometric congruence preserves area. Reversing the traversal would instead reverse the sign.

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