Tangentvektor

A tangentvektor, or tangent vector, is a vector tangent to a curve at a point or, more generally, an element of the tangent space of a smooth manifold at that point. In Euclidean space, if a curve is given by a differentiable parametrization c: I → R^n, the tangent vector at t0 is c′(t0). If c(t) = (x1(t), ..., xn(t)), then the tangent vector is (dx1/dt(t0), ..., dxn/dt(t0)). This vector represents the instantaneous direction and speed of motion along the curve.

The tangent vector is closely linked to velocity: for a point moving along c with parameter t, c′(t0) gives the velocity at t0. It also encapsulates directional derivatives: the rate of change of a function f in the direction of a tangent vector v is given by the directional derivative D_v f(p) = ∇f(p) · v in Euclidean space.

In the context of level sets, where a smooth function F: R^n → R vanishes on a curve or surface, tangent vectors at a point p satisfy ∇F(p) · v = 0. Thus they lie in the hyperplane orthogonal to the gradient, describing directions tangent to the level set at p.

More generally, for a smooth manifold M, the tangent space T_p M is the vector space of all tangent vectors at p. There are multiple equivalent constructions, such as velocity vectors of curves through p or derivations acting on smooth functions at p. The dimension of T_p M equals the dimension of M.

Example: for the circle r(t) = (cos t, sin t), the tangent vector is r′(t) = (−sin t, cos t), which is perpendicular to the radius at each point, i.e., lies along the circle’s tangent.