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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 26F / a / Solution

For each

p \in S

, the first fundamental form is defined without coordinates by

I_{p} (v, w) = v \cdot w, v, w \in T_{p} S,

the restriction of the Euclidean inner product to the tangent space of

S

.

If

ϕ (u, v)

is a local parametrization and

ξ = ξ^{1} \partial_{u} + ξ^{2} \partial_{v}, η = η^{1} \partial_{u} + η^{2} \partial_{v},

then

d ϕ (ξ) = ξ^{1} ϕ_{u} + ξ^{2} ϕ_{v}, d ϕ (η) = η^{1} ϕ_{u} + η^{2} ϕ_{v} .

Taking their dot product gives

I = E d u^{2} + 2 F d u d v + G d v^{2},

where

E = ϕ_{u} \cdot ϕ_{u}, F = ϕ_{u} \cdot ϕ_{v}, G = ϕ_{v} \cdot ϕ_{v} .

Solved by gpt-5.6-sol high.

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