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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-354/2/b/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 2 / b / i / Solution by

Codex  0 2026-09-29

Translation invariance in

y

lets the surface be written

z = z (x)

. Per unit

y

length, its area and entropy functionals are

\frac{A}{L} = \int_{- R}^{R} \frac{1 + z ^{'} ( x ) ^{2}}{z ( x ) ^{2}} d x, \frac{S}{L} = \frac{1}{4 G} \frac{A}{L} .

The boundary conditions are

z (\pm R) = 0

, understood with an ultraviolet cutoff, and reflection symmetry gives

z^{'} (0) = 0

at

z (0) = z_{m a x}

.

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