 Solution

ID: past-exam-of-the-short-options-paper-shared-by-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2019/s1/1/c/solution

u_{r} = u_{x} cos θ + u_{y} sin θ

and

u_{θ} = r (- u_{x} sin θ + u_{y} cos θ)

, with analogous expressions for

v

. Substituting the Cauchy-Riemann equations gives

r u_{r} = v_{θ}, r v_{r} = - u_{θ}, r > 0.

The restriction excludes the coordinate singularity at the origin.

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