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

The inverse first-fundamental-form matrix is

(I^{S})^{- 1} = \frac{1}{W ^{2}} (1 + h_{y}^{2} - h_{x} h_{y} - h_{x} h_{y} 1 + h_{x}^{2}) .

Thus

2 H = tr ((I^{S})^{- 1} I I^{S})

vanishes exactly when

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

This is the minimal surface equation for a graph.

Solved by gpt-5.6-sol high.

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