Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 15G (Geometry)

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- Solution 15G

Solution

Differentiate the parametrized surface. The coefficients of the first fundamental form simplify to

E = Q^{2} + \frac{v ^{2}}{4}, F = 0, G = 1.

If

c = σ_{u} \times σ_{v}

, then

∣ c ∣^{2} = E

. Moreover

σ_{uv} \cdot c = 1, σ_{vv} \cdot c = 0.

Thus the second-fundamental-form coefficients satisfy

M = 1/ E

and

N = 0

. The Gaussian curvature is consequently

K = \frac{L N - M ^{2}}{EG - F ^{2}} = - \frac{1}{E ^{2}} = - \frac{1}{( v ^{2} /4 + Q ^{2} ) ^{2}} .

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