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In mathematics, 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 linear 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. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure ''locally'' by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called ''transition maps.'' Differentiability means different things in different contexts including: continuously differentiable, ''k'' times differentiable, smooth, and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry. ==History== (詳細はCarl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture〔B. Riemann (1867).〕 before the faculty at Göttingen. He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments: : ''Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ...''– B. Riemann The works of physicists such as James Clerk Maxwell,〔Maxwell himself worked with quaternions rather than tensors, but his equations for electromagnetism were used as an early example of the tensor formalism; see .〕 and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita〔See G. Ricci (1888), G. Ricci and T. Levi-Civita (1901), T. Levi-Civita (1927).〕 led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformations. These ideas found a key application in Einstein's theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces.〔See H. Weyl (1955).〕 The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.〔H. Whitney (1936).〕 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「Differentiable manifold」の詳細全文を読む スポンサード リンク
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