 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2021/4/a/i/solution

Take

Ω = R^{2}

and

F (s, t) = (e^{s} cos t, e^{s} sin t) .

D F_{(s, t)} = e^{s} (cos t sin t - sin t cos t), det D F_{(s, t)} = e^{2 s} > 0.

F (s, t + 2 π) = F (s, t)

, so it is not injective and has no global inverse on its image. The domain is open and connected as required.

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