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in and
:

is a diffeomorphism. These local coordinates on give rise to an isomorphism for all . We may then define a map

:

by
:

We use these maps to define the topology and smooth structure on . A subset of is open if and only if

:

is open in for each These maps are homeomorphisms between open subsets of and and therefore serve as charts for the smooth structure on . The transition functions on chart overlaps are induced by the Jacobian matrices

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