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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 26J / e / Solution

The area element is

det I^{S} d x d y = W d x d y

, so

A (h) = \int_{Ω} 1 + h_{x}^{2} + h_{y}^{2} d x d y .

For the Lagrangian

L (p, q) = 1 + p^{2} + q^{2}

, stationarity under every compactly supported variation gives the Euler-Lagrange equation

\partial_{x} (\frac{h _{x}}{W}) + \partial_{y} (\frac{h _{y}}{W}) = 0.

Multiplication by

W^{3}

expands this into

(1 + h_{y}^{2}) h_{xx} - 2 h_{x} h_{y} h_{x y} + (1 + h_{x}^{2}) h_{yy} = 0,

recovering part (d).

Solved by gpt-5.6-sol high.

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