Manifold upon which it is possible to perform calculus
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A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector 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 vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their compositions on chart intersections in the atlas must be differentiable functions on the corresponding vector 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.
The ability to define such a local differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A locally 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.
"Differentiability" of a manifold has been given several meanings, including: continuously differentiable, k-times differentiable, smooth (which itself has many meanings), and analytic.
and 26 Related for: Differentiable manifold information
another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiablemanifold is a...
scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiablemanifolds; their differentiable structure...
applicable to general topological manifolds often employ methods of homology theory, whereas for differentiablemanifolds more structure is present, allowing...
are continuously differentiable. Given two differentiablemanifolds M {\displaystyle M} and N {\displaystyle N} , a differentiable map f : M → N {\displaystyle...
necessarily differentiable or piecewise differentiable. As above, let ( M , g ) {\displaystyle (M,g)} be a connected and continuous Riemannian manifold. The...
words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the...
mathematics, particularly differential geometry, a Finsler manifold is a differentiablemanifold M where a (possibly asymmetric) Minkowski norm F(x, −) is...
On a differentiablemanifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The...
In mathematics, a differentiablemanifold M {\displaystyle M} of dimension n is called parallelizable if there exist smooth vector fields { V 1 , … ,...
M as a Riemannian manifold can be recovered from this data. This suggests that one might define a noncommutative Riemannian manifold as a spectral triple...
{\displaystyle f:M\to N} from a differentiablemanifold M to another differentiablemanifold N gives rise to a differentiable fiber bundle. For one thing...
a field concerned more generally with geometric structures on differentiablemanifolds. A geometric structure is one which defines some notion of size...
\mathbf {Gr} _{k}(V)} (named in honour of Hermann Grassmann) is a differentiablemanifold that parameterizes the set of all k {\displaystyle k} -dimensional...
form is a differential form of degree equal to the differentiablemanifold dimension. Thus on a manifold M {\displaystyle M} of dimension n {\displaystyle...
different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiablemanifolds. For example, the Whitney embedding...
surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiablemanifold with a connection. It is a generalization...
numbers.[clarification needed] In other words, a differentiable curve is a differentiablemanifold of dimension one. In Euclidean geometry, an arc (symbol:...
In mathematics, an analytic manifold, also known as a C ω {\displaystyle C^{\omega }} manifold, is a differentiablemanifold with analytic transition maps...
of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function...
Hamiltonian field theory. Mathematics portal Almost symplectic manifold – differentiablemanifold equipped with a nondegenerate (but not necessarily closed)...
forgetful functors: for instance, a differentiablemanifold is also a topological manifold, and a differentiable map is also continuous, so there is a...
tangent vector). More generally, vector fields are defined on differentiablemanifolds, which are spaces that look like Euclidean space on small scales...
constraints can be generalized to finding local maxima and minima on a differentiablemanifold M . {\displaystyle \ M~.} In what follows, it is not necessary...
Andrew Wallace called it spherical modification. The "surgery" on a differentiablemanifold M of dimension n = p + q + 1 {\displaystyle n=p+q+1} , could be...
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