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

Table of contents
- Solution c

Solution

Write

D = 1 + ψ_{r}^{2} + ψ_{z}^{2} .

Differentiating the prescribed identity

u (ψ (r, z), r, z) = 0

in its two tangential directions gives

u_{r} = - ψ_{r} u_{t}, u_{z} = - ψ_{z} u_{t} .

The unit normal is

(1, - ψ_{r}, - ψ_{z}) / D

, so the second item of Cauchy data becomes

g = \partial_{N} u = \frac{u _{t} - ψ _{r} u _{r} - ψ _{z} u _{z}}{D} = D u_{t} .

Consequently

u_{t} = \frac{g}{D}, u_{r} = - \frac{ψ _{r} g}{D}, u_{z} = - \frac{ψ _{z} g}{D} .

Substitution into the principal symbol from part a shows that the graph is non-characteristic exactly where

1 - (1 - \frac{ψ _{z} g}{1 + ψ _{r}^{2} + ψ _{z}^{2}})^{2} (ψ_{r}^{2} + ψ_{z}^{2}) \neq = 0.

Solved by gpt-5.6-sol high.

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