 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/f/f2/a/ii/solution

f : U \to R^{n}

C^{1}

on an open set and

D f (a)

U_{0}

a

and

V_{0}

f (a)

such that

f : U_{0} \to V_{0}

is a

C^{1}

diffeomorphism. Its inverse satisfies

D (f^{- 1}) (f (x)) = (D f (x))^{- 1}, x \in U_{0} .

f

C^{k}

or smooth, the inverse has the same regularity.

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