Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 105 / 1 / a

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- Solution a

Solution

Only the second-order terms contribute to the principal symbol. For a covector

ξ = (ξ_{t}, ξ_{r}, ξ_{z})

it is

p (ξ) = ξ_{t}^{2} - (1 + u_{z})^{2} (ξ_{r}^{2} + ξ_{z}^{2}) .

The conormal to the hypersurface

Σ = {ϕ = 0}

is

d ϕ = (ϕ_{t}, ϕ_{r}, ϕ_{z})

. After evaluating the coefficient at the prescribed boundary value of

u_{z}

, the non-characteristic condition is therefore

ϕ_{t}^{2} - (1 + u_{z} ∣_{Σ})^{2} (ϕ_{r}^{2} + ϕ_{z}^{2}) \neq = 0.

Solved by gpt-5.6-sol high.

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