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- If two charts are smoothly compatible with an atlas then they are smothly compatible with each other...oth manifold]] from a [[topological manifold]]. We may speak of a (smooth) atlas on a space that is simply locally euclidean, it need not be a full on [[top *# {{M|(V,\psi)}} is [[smoothly compatible with the atlas]] {{M|\mathcal{A} }} and6 KB (1,182 words) - 13:38, 1 April 2017
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- ...on map]] for moving between charts, and [[Smoothly compatible charts]] for the smooth form. * {{M|U}} is called the ''coordinate domain'' or ''coordinate neighbourhood'' of each of its points2 KB (322 words) - 06:32, 7 April 2015
- ...t|charts]], {{M|(U,\varphi)}} and {{M|(V,\psi)}} are said to be ''smoothly compatible''<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> This is vital to define [[Smooth atlas|smooth atlases]]727 B (101 words) - 12:27, 12 November 2015
- '''Note:''' a smooth atlas is a special kind of [[Atlas]] An [[Atlas|atlas]] {{M|\mathcal{A} }} is called a ''smooth atlas''<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref>2 KB (242 words) - 12:27, 12 November 2015
- ===Smoothly compatible charts=== ...arts|smoothly compatible charts]]) - Two charts are smoothly compatible if the intersections of their domains is empty, or there is a [[Diffeomorphism|dif3 KB (413 words) - 21:09, 12 April 2015
- ...as, or a smooth atlas which is not properly contained in any larger smooth atlas ...atlas. However as this example shows there are many smooth atlases giving the same "smooth structure"2 KB (246 words) - 07:10, 7 April 2015
- ** That is to say {{M|f\circ\varphi^{-1} }} is [[Smooth|smooth]] in the usual sense - of having continuous partial derivatives of all orders. Theorem: Any other chart in {{M|(M,\mathcal{A})}} will also satisfy the definition of {{M|f}} being smooth3 KB (560 words) - 16:16, 14 April 2015
- ...function {{M|f:M\rightarrow\mathbb{R} }} which is continuous. Notice that the notion of continuity on such a map is easy! However suppose we want to diff ...hbb{R}^n}}", {{M|\psi}} and {{M|\varphi}} are [[chart|charts]] from some [[atlas]].2 KB (414 words) - 12:26, 12 November 2015
- If two charts are smoothly compatible with an atlas then they are smothly compatible with each other...oth manifold]] from a [[topological manifold]]. We may speak of a (smooth) atlas on a space that is simply locally euclidean, it need not be a full on [[top *# {{M|(V,\psi)}} is [[smoothly compatible with the atlas]] {{M|\mathcal{A} }} and6 KB (1,182 words) - 13:38, 1 April 2017