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- Differentiable_manifold abstract "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, 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.".
- Differentiable_manifold thumbnail Nondifferentiable_atlas.png?width=300.
- Differentiable_manifold wikiPageExternalLink SmoothManifold.html.
- Differentiable_manifold wikiPageExternalLink Geom.
- Differentiable_manifold wikiPageID "2119219".
- Differentiable_manifold wikiPageRevisionID "605371903".
- Differentiable_manifold b "p".
- Differentiable_manifold caption "Charts on a manifold".
- Differentiable_manifold hasPhotoCollection Differentiable_manifold.
- Differentiable_manifold id "p/d031790".
- Differentiable_manifold imageWidth "250".
- Differentiable_manifold p "k".
- Differentiable_manifold title "Differentiable manifold".
- Differentiable_manifold subject Category:Smooth_manifolds.
- Differentiable_manifold type Artifact100021939.
- Differentiable_manifold type Conduit103089014.
- Differentiable_manifold type Manifold103717750.
- Differentiable_manifold type Object100002684.
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- Differentiable_manifold type SmoothManifolds.
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- Differentiable_manifold type YagoPermanentlyLocatedEntity.
- Differentiable_manifold comment "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.".
- Differentiable_manifold label "Differentiable manifold".
- Differentiable_manifold label "Differentieerbare variëteit".
- Differentiable_manifold label "Differenzierbare Mannigfaltigkeit".
- Differentiable_manifold label "Rozmaitość różniczkowa".
- Differentiable_manifold label "Variedad diferenciable".
- Differentiable_manifold label "Varietà differenziabile".
- Differentiable_manifold label "Variété différentielle".
- Differentiable_manifold label "Гладкое многообразие".
- Differentiable_manifold label "微分流形".
- Differentiable_manifold sameAs Differenzierbare_Mannigfaltigkeit.
- Differentiable_manifold sameAs Variedad_diferenciable.
- Differentiable_manifold sameAs Variété_différentielle.
- Differentiable_manifold sameAs Varietà_differenziabile.
- Differentiable_manifold sameAs Differentieerbare_variëteit.
- Differentiable_manifold sameAs Rozmaitość_różniczkowa.
- Differentiable_manifold sameAs m.06ngpp.
- Differentiable_manifold sameAs Q3552958.
- Differentiable_manifold sameAs Q3552958.
- Differentiable_manifold sameAs Differentiable_manifold.
- Differentiable_manifold wasDerivedFrom Differentiable_manifold?oldid=605371903.
- Differentiable_manifold depiction Nondifferentiable_atlas.png.
- Differentiable_manifold isPrimaryTopicOf Differentiable_manifold.