 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-115/2/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

(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} .

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