 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/16/a/i/solution

The logarithm is defined away from the

z

axis, where

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

. For the vector potential

ψ e_{z} = (0, 0, ψ)

, its curl is

(ψ_{y}, - ψ_{x}, 0)

. Since

ψ_{x} = \frac{a x}{x ^{2} + y ^{2}}, ψ_{y} = \frac{a y}{x ^{2} + y ^{2}},

we obtain

u = (\frac{a y}{x ^{2} + y ^{2}}, - \frac{a x}{x ^{2} + y ^{2}}, 0), x^{2} + y^{2} > 0.

For

a > 0

this is clockwise circulation about the axis, the velocity structure of a point vortex in each transverse plane.

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