 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1/1/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Along the circle with increasing

ϕ

d r = a sin α e_{ϕ} d ϕ

. Thus the two circulations are

\oint_{C} A^{(+)} \cdot d r = 2 π sin α tan (α /2), \oint_{C} A^{(-)} \cdot d r = - 2 π sin α cot (α /2) .

\oint_{C} A^{(+)} \cdot d r = 2 π (1 - cos α), \oint_{C} A^{(-)} \cdot d r = - 2 π (1 + cos α) .

Here

0 < α < π

so both potentials are defined on the whole curve.

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