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 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,
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
vector fields. Differentiable manifolds are very important in
physics. Special kinds of differentiable manifolds form the basis for physical theories such as
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