 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-1/5c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Put

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

. On the specified domain,

r > 0

\nabla \cdot u = U^{'} (z), \nabla \times u = 0 .

For example, the third component of the curl is

A (r^{2} - 2 x^{2}) / r^{4} + A (r^{2} - 2 y^{2}) / r^{4} = 0

; the other two components vanish because

U

depends only on

z

and the transverse velocity is independent of

z

. Hence the flow is both incompressible and irrotational exactly when

U (z) = U_{0} is constant, A is arbitrary .

The exclusion of the axis matters: these are ordinary derivatives on the domain, with no assertion of regularity on the removed axis.

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