 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-2/11/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

u > 0

\partial_{x} = cos v \partial_{u} - (sin v / u) \partial_{v}

and

\partial_{y} = sin v \partial_{u} + (cos v / u) \partial_{v}

. Write

c = cos v

s = sin v

. Applying these operators twice, including their variable coefficients, gives the Cartesian second derivatives in polar coordinates:

f_{xx} = c^{2} f_{uu} - \frac{2 sc}{u} f_{uv} + \frac{s ^{2}}{u ^{2}} f_{vv} + \frac{s ^{2}}{u} f_{u} + \frac{2 sc}{u ^{2}} f_{v},

f_{yy} = s^{2} f_{uu} + \frac{2 sc}{u} f_{uv} + \frac{c ^{2}}{u ^{2}} f_{vv} + \frac{c ^{2}}{u} f_{u} - \frac{2 sc}{u ^{2}} f_{v} .

f_{uu} + f_{u} / u + f_{vv} / u^{2}

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