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

Solution

(x^{1}, \dots, x^{n})

d vol_{g} = det (g_{ij}) d x^{1} \land \dots \land d x^{n} .

∣ g ∣ = det (g_{ij})

and

(g^{ij}) = (g_{ij})^{- 1}

f

E (u) = \int_{M} (\frac{1}{2} g^{ij} (x) \partial_{i} u \partial_{j} u - f u) ∣ g ∣ d x^{1} \dots d x^{n} .

Solved by gpt-5.6-sol high.

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