Tangent bundle
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).