tangentvektoren

Tangentvektoren, or tangent vectors, are mathematical objects associated with differentiable manifolds. A tangent vector at a point p on a manifold M can be defined in two equivalent ways. One common definition treats a tangent vector v at p as a linear map from the space C∞(M) of smooth real-valued functions defined near p to R that satisfies the Leibniz rule v(fg) = f(p) v(g) + g(p) v(f). Such a map encodes the first-order, directional rate of change of functions at p. Equivalently, a tangent vector is the velocity at t = 0 of a smooth curve γ: (−ε, ε) → M with γ(0) = p, via v(f) = d/dt|_0 f(γ(t)).

The collection of all tangent vectors at p forms the tangent space T_p M, a real vector space whose dimension equals dim M. In a local chart (U, x^1, ..., x^n) around p, the vectors ∂/∂x^i|_p form a basis, and every v ∈ T_p M can be written as v = ∑_i v^i ∂/∂x^i|_p.

The tangent bundle TM is the disjoint union of the tangent spaces over all points of M, with a natural projection τ: TM → M sending a vector to its base point. Each fiber T_p M is a copy of the tangent space at p, and TM has a smooth structure of a manifold of dimension 2 dim M.

In Euclidean space M = R^n, T_p M can be identified with R^n, and a tangent vector corresponds to the velocity vector of a curve through p. Tangent vectors act on smooth functions as directional derivatives: v(f) equals the directional derivative of f at p in the direction determined by v. Two curves that agree up to first order at p determine the same tangent vector.