 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/b/solution

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

f_{xx} = f_{uu} u_{x}^{2} + 2 f_{uv} u_{x} v_{x} + f_{vv} v_{x}^{2} + f_{u} u_{xx} + f_{v} v_{xx},

and

f_{yy} = f_{uu} u_{y}^{2} + 2 f_{uv} u_{y} v_{y} + f_{vv} v_{y}^{2} + f_{u} u_{yy} + f_{v} v_{yy} .

The last two terms account for derivatives of the coordinate functions; they vanish for an affine map. Smoothness justifies equality of the two mixed partial derivatives.

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