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ID: past-exam-of-the-options-i-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/f/1/c/i/solution

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Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / F / 1 / c / i / Solution by

Codex  0 2026-10-31

C^{1}

diffeomorphism

F : U \to V

is a bijection for which both

F

and

F^{- 1}

are continuously differentiable. If a higher regularity class is specified, both maps must belong to that class. In particular, a diffeomorphism is a homeomorphism, and the chain rule gives

D (F^{- 1}) (F (x)) = (D F (x))^{- 1}

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