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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 2G / Solution

Use the global parametrization

X (u, v) = (u, v, h (u, v)) .

Its tangent vectors are

X_{u} = (1, 0, h_{u}), X_{v} = (0, 1, h_{v}),

and

X_{u} \times X_{v} = (- h_{u}, - h_{v}, 1) \neq = 0.

Thus

X

is a regular parametrization and the graph is a smooth surface.

The first fundamental form is

I = (1 + h_{u}^{2}) d u^{2} + 2 h_{u} h_{v} d u d v + (1 + h_{v}^{2}) d v^{2} .

Choosing the upward orientation, the Gauss map is

N (u, v) = \frac{( - h _{u} , - h _{v} , 1 )}{1 + h _{u}^{2} + h _{v}^{2}} .

Solved by gpt-5.6-sol high.

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