 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-v/3/b/solution

Write

r = x^{2} + y^{2} > 0

, so

F = (f (z) x / r, f (z) y / r, 0)

. The Cartesian curl is

\nabla \times F = (- f^{'} (z) \frac{y}{r}, f^{'} (z) \frac{x}{r}, \partial_{x} \frac{f ( z ) y}{r} - \partial_{y} \frac{f ( z ) x}{r}) .

The last component is zero because both differentiated terms are

- f (z) x y / r^{3}

. Thus

\nabla \times F = f^{'} (z) (- sin θ, cos θ, 0) = f^{'} (z) e_{θ} .

The condition

r > 0

excludes the cylindrical-axis coordinate singularity.

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