tangentsystem

The tangentsystem is a concept used in mathematics, particularly in differential geometry, to refer to the system of tangent vectors attached to points of a space or to the space of all such vectors. In English-language literature the term is uncommon; more standard terminology is tangent space at a point or tangent bundle. The expression is sometimes encountered in German-language texts (Tangentsystem) or used descriptively to describe the collection of tangent directions across a manifold.

Formally, for a smooth manifold M, the tangent space at a point p ∈ M, denoted T_pM, is a real vector space that captures the possible directions in which one can move from p within M. T_pM can be defined in several equivalent ways, for example as velocity vectors of smooth curves passing through p or as derivations at p on the algebra of smooth functions. The tangentsystem at p is this vector space T_pM. The full tangential structure of M is organized into the tangent bundle TM = ⋃_{p∈M} T_pM, whose points are pairs (p, v) with v ∈ T_pM.

Examples help illustrate the idea: for M = R^n, T_pR^n ≅ R^n, and the tangent vectors are simply velocity vectors of curves through p. If γ(t) is a curve in M with γ(0) = p, then γ′(0) ∈ T_pM. For a surface embedded in R^3, the tangent space at a point is the plane spanned by the partial derivative directions in a local chart.

Applications of the tangentsystem include the linear approximation of manifolds, the definition of vector fields and flows, and the formulation of physical theories and geometric analysis. Related concepts are tangent space, tangent bundle, and the differential (pushforward) between manifolds.