The tangent plane to a surface at a point p is the tangent space at p (after translating to the origin). The elements of the tangent space are called tangent vectors, and they are closed under addition and scalar multiplication. In particular, the tangent space is a vector space. Any submanifold of Euclidean space, and more generally any submanifold of an abstract manifold, has a tangent space at each point. The collection of tangent spaces T M_p to M forms the tangent bundle T M = union _(p element M)(p, T M_p). A vector field assigns to every point p a tangent vector in the tangent space at p.

calculus | chart tangent space | coordinate chart | differential k-form | directional derivative | Euclidean space | exterior algebra | intrinsic tangent space | Jacobian | linear algebra | manifold | null space | tangent bundle | tangent plane | tangent vector | vector field | vector space | velocity vector