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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 39D / iii / Solution

Taking the trace of the velocity gradient gives

\nabla \cdot u = s (\frac{A ^{'}}{r} + r B^{'} + 4 B) .

Direct substitution yields

\frac{A ^{'}}{r} + r B^{'} + 4 B = 0,

so

\nabla \cdot u = 0.

This verifies incompressibility directly.

Solved by gpt-5.6-sol high.

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