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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/20/c/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 20 / c / Solution by

Codex  0 2026-10-18

The curl gives the vorticity:

\nabla \times v = (- \partial_{z} (αx z), \partial_{z} (- α yz), \partial_{x} (αx z) - \partial_{y} (- α yz)) = (- αx, - α y, 2 α z) .

(1)

Since

x i + y j = ρ ρ

, the requested form is

\nabla \times v = - α ρ ρ + 2 α z k .

(2)

At the axis the radial unit vector is undefined, but its coefficient vanishes and the Cartesian expression remains smooth. The radial vorticity reflects the change in rotation rate with height.

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