 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/3/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Use the chain rule with

x = r cos θ

y = r sin θ

. Then

U_{r} = u_{x} cos θ + u_{y} sin θ

and

V_{θ} = - r v_{x} sin θ + r v_{y} cos θ

. Substituting the Cartesian Cauchy-Riemann equations gives

V_{θ} = r U_{r}

. Similarly

V_{r} = v_{x} cos θ + v_{y} sin θ = - U_{θ} / r

. Thus, for

r > 0

, the polar equations are

U_{r} = \frac{1}{r} V_{θ}, V_{r} = - \frac{1}{r} U_{θ} .

The restriction

r > 0

is needed because the polar chart degenerates at the origin.

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