Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 12 / b / Solution

Differentiate the angular relation locally, or use

d ϕ = (x d y - y d x) / (x^{2} + y^{2})

. Holding each other coordinate fixed gives

ϕ_{x} = - \frac{y}{r ^{2}} = - \frac{sin ϕ}{r}, ϕ_{y} = \frac{x}{r ^{2}} = \frac{cos ϕ}{r} .

These are local partial derivatives on a consistent angular branch, independent of where its branch cut is placed.

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