 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/13/a/ii/solution

Parametrize the circle by

r (t) = (cos t, sin t)

0 \leq t \leq 2 π

. Its tangent is

r^{'} = (- sin t, cos t) = F (r (t))

F \cdot r^{'}

is one. For the other field,

G \cdot r^{'} = 2 sin t cos t (cos^{2} t - sin^{2} t) = \frac{1}{2} sin 4 t .

Integrating around the full circle gives

\oint_{circle} F \cdot d r = 2 π, \oint_{circle} G \cdot d r = 0.

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