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A Course in Mathematical Analysis by D. J. H. Garling

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19

Differential manifolds in Euclidean space

19.1 Differential manifolds in Euclidean space

A manifold is a topological space which is locally like Euclidean space: each point has an open neighbourhood which is homeomorphic to an open subset of a Euclidean space. A differential manifold is one for which the homeomorphisms can be taken to be diffeomorphisms. We consider differential manifolds which are subspaces of Euclidean space.

Recall that a diffeomorphism f of an open subset W of a Euclidean space E onto a subset f(W) of a Euclidean space F is a bijection of W onto f(W) which is continuously differentiable, and has the property that the derivative Dfx is invertible, for each xW. If so, then f(W) is open in F, and the mapping f1 : f

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