 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/7/a/solution

z = r e^{i θ}

, the chain rule gives

f_{r} = e^{i θ} f^{'} (z), f_{θ} = i r e^{i θ} f^{'} (z) = i r f_{r} .

f = u + i v

u_{θ} = - r v_{r}, v_{θ} = r u_{r} .

For

r > 0

r

\frac{1}{r} u_{θ} = - v_{r}, u_{r} = \frac{1}{r} v_{θ} .

The restriction

r > 0

reflects the coordinate singularity at the origin; the Cartesian equations have no such restriction.

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