Differentiable manifold | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-04

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Differentiable_manifold


00:02:39 1 History
00:04:17 2 Definition
00:08:14 2.1 Atlases
00:10:53 2.2 Compatible atlases
00:11:52 3 Alternative definitions
00:12:02 3.1 Pseudogroups
00:14:51 3.2 Structure sheaf
00:15:08 3.2.1 Sheaves of local rings
00:16:27 4 Differentiable functions
00:16:56 4.1 Differentiation of functions
00:21:13 4.1.1 Directional differentiation
00:23:43 4.1.2 Tangent vector and the differential
00:24:20 4.2 Partitions of unity
00:25:23 4.3 Differentiability of mappings between manifolds
00:25:48 4.4 Algebra of scalars
00:26:54 5 Bundles
00:27:36 5.1 Tangent bundle
00:30:14 5.2 Cotangent bundle
00:32:21 5.3 Tensor bundle
00:33:40 5.4 Frame bundle
00:35:59 5.5 Jet bundles
00:36:08 6 Calculus on manifolds
00:37:40 6.1 Differential calculus of functions
00:39:04 6.2 Lie derivative
00:39:59 6.3 Exterior calculus
00:40:47 6.3.1 Exterior derivative
00:42:27 7 Topology of differentiable manifolds
00:44:44 7.1 Relationship with topological manifolds
00:47:54 7.2 Classification
00:49:06 8 Structures on manifolds
00:50:02 8.1 (Pseudo-)Riemannian manifolds
00:52:46 8.2 Symplectic manifolds
00:52:57 8.3 Lie groups
00:53:58 9 Generalizations
00:56:36 10 See also
00:56:46 11 Notes
00:58:09 12 References
00:59:00 13 Bibliography
00:59:24 Generalizations
01:00:19 See also



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SUMMARY
=======
In mathematics, a differentiable manifold (also differential 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.