 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/12/c/solution

Apply the multivariable chain rule to

f (x, y) = g (r (x, y), ϕ (x, y))

. Using the coordinate partial derivatives above gives

f_{x} = g_{r} cos ϕ - \frac{g _{ϕ}}{r} sin ϕ, f_{y} = g_{r} sin ϕ + \frac{g _{ϕ}}{r} cos ϕ .

Equivalently,

\partial_{x} = cos ϕ \partial_{r} - (sin ϕ / r) \partial_{ϕ}

and

\partial_{y} = sin ϕ \partial_{r} + (cos ϕ / r) \partial_{ϕ}

. Their coefficients vary with position and must also be differentiated when applying them twice.

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