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

Write

h_{x} = \partial_{x} h

and

h_{y} = \partial_{y} h

. Since

ϕ_{x} = (1, 0, h_{x}), ϕ_{y} = (0, 1, h_{y}),

the first fundamental form in the coordinate basis is

I^{S} = (1 + h_{x}^{2} h_{x} h_{y} h_{x} h_{y} 1 + h_{y}^{2}) .

Solved by gpt-5.6-sol high.

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