 Solution

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/a/solution

Split the vector field into

a (- y, x, 0)

and

e^{- r^{2}} (x, y, z)

. The second part is the gradient of

- \frac{1}{2} e^{- r^{2}}

, so its curl vanishes. The first part has only a vertical curl component,

\partial_{x} (a x) - \partial_{y} (- a y) = 2 a

. Hence

\nabla \times F = (0, 0, 2 a) .

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