 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-3/7c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Choose the polar axis along

d

and write

d = ∣ d ∣

. Since

d = d (cos θ e_{r} - sin θ e_{θ}),

the velocity components are

u_{r} = \frac{d c o s θ}{r ^{2}}, u_{θ} = \frac{d s i n θ}{r ^{2}} .

\nabla \cdot u = \frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) + \frac{1}{r} \frac{\partial u _{θ}}{\partial θ} = - \frac{d c o s θ}{r ^{3}} + \frac{d c o s θ}{r ^{3}} = 0.

Thus the flow is incompressible for $r \neq = 0$ .

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