Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 332 / 4 / b

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- Solution b

Solution

For the surface

F (x, z, t) = z - ζ (x, t) = 0

, an outward normal vector and positively oriented tangent vector are

\hat{n} = \frac{( - ζ _{x} , 1 )}{1 + ζ _{x}^{2}}, \hat{t} = \frac{( 1 , ζ _{x} )}{1 + ζ _{x}^{2}} .

With

ζ = ϵ e^{ik x}

and

∣ k ϵ ∣ ≪ 1

, linearization gives

\hat{n} = (- ik ϵ e^{ik x}, 1) + O (ϵ^{2} k^{2}), \hat{t} = (1, ik ϵ e^{ik x}) + O (ϵ^{2} k^{2}) .

Solved by gpt-5.6-sol high.

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