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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Differential structure ： ウィキペディア英語版 Differential structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If ''M'' is already a topological manifold, it is required that the new topology be identical to the existing one. ==Definition== For a natural number ''n'' and some ''k'' which may be a non-negative integer or infinity, an n-dimensional ''Ck'' differential structure 〔Hirsch, Morris, ''Differential Topology'', Springer (1997), ISBN 0-387-90148-5. for a general mathematical account of differential structures〕 is defined using a Ck-atlas, which is a set of bijections called charts between a collection of subsets of ''M'' (whose union is the whole of ''M''), and a set of open subsets of $\backslash mathbb^$: :$\backslash varphi\_:M\backslash supset\; W\_\backslash rightarrow\; U\_\backslash subset\backslash mathbb^$ which are ''Ck''-compatible (in the sense defined below): Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of $\backslash mathbb^$ but the usefulness of this notion depends on to what extent these notions agree when the domains of two such maps overlap. Consider two charts: :$\backslash varphi\_:W\_\backslash rightarrow\; U\_,\backslash ,$ :$\backslash varphi\_:W\_\backslash rightarrow\; U\_.\backslash ,$ The intersection of the domains of these two functions is: :$W\_=W\_\backslash cap\; W\_\backslash ;$ and its map by the two chart maps to the two images: :$U\_=\backslash varphi\_\backslash left(W\_\backslash right),\backslash ,$ :$U\_=\backslash varphi\_\backslash left(W\_\backslash right)$ The transition map between the two charts is the map between the two images of this intersection under the two chart maps. :$\backslash varphi\_:U\_\backslash rightarrow\; U\_$ :$\backslash varphi\_(x)=\backslash varphi\_\backslash left(\backslash varphi\_^\backslash left(x\backslash right)\backslash right).$ Two charts $\backslash varphi\_,\backslash ,\backslash varphi\_$ are Ck-compatible if :$U\_,\backslash ,\; U\_$ are open, and the transition maps :$\backslash varphi\_,\backslash ,\backslash varphi\_$ have continuous derivatives of order ''k''. If ''k = 0'', we only require that the transition maps are continuous, consequently a ''C0''-atlas is simply another way to define a topological manifold. If ''k'' = ∞, derivatives of all orders must be continuous. A family of ''Ck''-compatible charts covering the whole manifold is a ''Ck''-atlas defining a ''Ck'' differential manifold. Two atlases are ''Ck''-equivalent if the union of their sets of charts forms a ''Ck''-atlas. In particular, a ''Ck''-atlas that is ''C0''-compatible with a ''C0''-atlas that defines a topological manifold is said to determine a ''Ck'' differential structure on the topological manifold. The ''Ck'' equivalence classes of such atlases are the distinct Ck differential structures of the manifold. Each distinct differential structure is determined by a unique maximal atlas, which is simply the union of all atlases in the equivalence class. For simplification of language, without any loss of precision, one might just call a maximal ''C''''k''−atlas on a given set a ''C''''k''−manifold. This maximal atlas then uniquely determines both the topology and the underlying set, the latter being the union of the domains of all charts, and the former having the set of all these domains as a basis. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Differential structure」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.