 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/13/a/solution

\nabla \times F = (0, - x y sinh z, x cosh z),

which is not identically zero. Thus $F$ is not conservative on

R^{3}

and cannot equal a negative gradient there.

G = \nabla (x^{2} y cosh z) .

Therefore $G$ is conservative, and the requested negative-gradient potential is

Φ = - x^{2} y cosh z + C, G = - \nablaΦ.

Its curl vanishes, consistently with the curl of a gradient identity.

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