 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/37d/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

x = r cos θ

and

y = r sin θ

. The chain rule gives

\partial_{r} = cos θ \partial_{x} + sin θ \partial_{y}, \partial_{θ} = - r sin θ \partial_{x} + r cos θ \partial_{y} .

∥ \partial_{r} ∥ = 1

and

∥ \partial_{θ} ∥ = r

for

r > 0

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