Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 2 / b

Table of contents
- Solution b

Solution

For an arbitrary smooth variation

u_{t} = u + tφ

, differentiation under the integral gives

\frac{d}{d t} E (u_{t})_{t = 0} = \int_{M} (g^{ij} \partial_{i} u \partial_{j} φ - f φ) ∣ g ∣ d x .

Since

M

is compact without boundary, integration by parts turns this into

- \int_{M} [\frac{1}{∣ g ∣} \partial_{j} (∣ g ∣ g^{ij} \partial_{i} u) + f] φ d vol_{g} .

The fundamental lemma of the calculus of variations therefore gives the Euler-Lagrange equation

- \frac{1}{∣ g ∣} \partial_{j} (∣ g ∣ g^{ij} \partial_{i} u) = f,

or equivalently

- Δ_{g} u = f

for the Laplace-Beltrami operator.

Solved by gpt-5.6-sol high.

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