Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 4 / b

Table of contents
- Solution b

Solution

Let

\overline{\nabla} f

and

\overline{Hess} f

be the ambient gradient and Hessian. The Laplacian of a restricted ambient function is

Δ_{M} (f ∣_{M}) = tr_{TM} (\overline{Hess} f) + ⟨ \overline{\nabla} f, H ⟩ .

To prove it, choose a local orthonormal tangent frame

e_{1}, \dots, e_{n}

with

\nabla_{e_{i}}^{M} e_{j} = 0

at the point under consideration. There,

Δ_{M} (f ∣_{M}) = i \sum e_{i} (e_{i} f) = i \sum (\overline{Hess} f (e_{i}, e_{i}) + ⟨ \overline{\nabla} f, \overline{\nabla}_{e_{i}} e_{i} ⟩) = i \sum \overline{Hess} f (e_{i}, e_{i}) + ⟨ \overline{\nabla} f, i \sum A (e_{i}, e_{i}) ⟩ .

Both sides are intrinsic scalars, so the pointwise calculation proves the formula everywhere.

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