Tangent bundle Guide, Meaning , Facts, Information and Description
In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure.The tangent bundle of manifold M\ is usually denoted by T(M) or just TM. Any element of T(M) is a pair (x,v) where v ∈ Tx(M), the tangent space at x. If M is n-dimensional and φ : U → Rn is a coordinate chart then the preimage V of U in T(M) admits a map to ψ : V → Rn × Rn defined by ψ(x, v) = (φ(x), dφ(v)). This map is taken to be a chart (by definition) and it defines structure of smooth 2n-dimensional manifold on T(M).
See also: Cotangent bundle