 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/13/b/solution

\nabla \times \nablaΦ = (Φ_{zy} - Φ_{yz}, Φ_{x z} - Φ_{z x}, Φ_{y x} - Φ_{x y}) .

Equality of mixed partial derivatives makes every component zero. Thus

\nabla \times F = 0 .

This is the curl of a gradient identity.

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