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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/2021/paper-2/16/a/ii/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 / 2021 / Paper 2 / 16 / a / ii / Solution by

Codex  0 2026-10-18

On the same domain, differentiate the Cartesian components:

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

(1)

Thus

\nabla \cdot u = 0 (x^{2} + y^{2} > 0) .

(2)

This is also an instance of divergence of a curl is zero, since the potential is smooth off the axis. The classical field is undefined on the axis, so that identity must not be applied there without specifying an extension.

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