In topology, a tangent bundle of a given manifold is a new manifold that consists of the tangent spaces for each point pasted together in a continuous fashion.

Every smooth manifold M has a tangent bundle T M, which consists of the tangent space T M_p at all points p in M. Since a tangent space T M_p is the set of all tangent vectors to M at p, the tangent bundle is the collection of all tangent vectors, along with the information of the point to which they are tangent.

calculus | coordinate chart | cotangent bundle | directional derivative | Euclidean space | Jacobian | manifold | tangent bundle section | tangent space | tangent vector | vector field | vector space

graduate school level