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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 25J / c / Solution

Apply change of variables to the orientation-preserving diffeomorphism

ψ_{ε} : A \to A

:

vol (A) = \int_{A} det D ψ_{ε} = (1 + ε^{2})^{- (n + 1) /2} \int_{A} det (I + ε D ϕ) .

Rearranging gives

\int_{A} det (I + ε D ϕ) = vol (A) (1 + ε^{2})^{(n + 1) /2} .

Solved by gpt-5.6-sol high.

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