Differentiable manifold

The notation of  a differentiable manifold refines that a manifold by requiring the functions that transform between charts to be differentiable. A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a liner space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

Atlas

One describes a manifold using an atlas.

Defintion An atlas for a topological spae M is a collection \{(U_\alpha,\varphi_\alpha)\} of charts on M such that \bigcup U_\alpha=M.

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