Why define tangent vectors as derivations?

The derivation definition is intrinsic and coordinate-free: a tangent vector is a linear operator on smooth functions satisfying the Leibniz rule, which avoids reference to any embedding and works on abstract manifolds.

What does the Lie bracket of two vector fields measure?

It measures the failure of the flows of the two vector fields to commute; vanishing of the bracket means the flows can be followed in either order to reach the same point.

Tangent Spaces and Vector Fields

The tangent space attaches a vector space of velocities to each point of a manifold, and a vector field assigns such a velocity smoothly across the manifold, encoding flows and infinitesimal symmetries.

Definition

The tangent space at a point of a smooth manifold is the vector space of velocity vectors of curves through that point (equivalently, derivations of smooth functions at the point); a vector field is a smooth assignment of a tangent vector to each point, i.e. a section of the tangent bundle.

Scope

This topic defines the tangent space — equivalently via velocity vectors of curves, derivations, or transition-compatible tuples — and assembles tangent spaces into the tangent bundle. It develops the differential of a smooth map, vector fields as sections of the tangent bundle, their integral curves and flows, the Lie bracket and Lie derivative, and Frobenius' theorem on integrability of distributions. Cotangent spaces and one-forms appear as the dual structure leading toward differential forms.

Core questions

What are the equivalent definitions of a tangent vector, and why do they agree?
How does the differential of a smooth map act on tangent spaces?
How do vector fields generate flows, and what does the Lie bracket measure about two flows?
When can a family of tangent distributions be integrated into submanifolds (Frobenius' theorem)?

Key concepts

Tangent space and tangent vectors as derivations
Tangent bundle and the differential of a smooth map
Vector fields, integral curves, and flows
Lie bracket and Lie derivative
Distributions and the Frobenius integrability theorem

Clinical relevance

Tangent vectors and vector fields formalize velocity, force, and infinitesimal symmetry; they are the substrate for dynamical systems on manifolds, the Lie algebra of a Lie group, and the geodesic and curvature constructions of Riemannian geometry.

History

The intrinsic, coordinate-free definition of the tangent space as derivations emerged in the mid-20th century, building on Lie's theory of continuous transformation groups and Cartan's calculus of differential forms, giving differential geometry its modern functorial formulation.

Key figures

Élie Cartan
Sophus Lie
John M. Lee

Seminal works

lee2012
warner1983

Frequently asked questions

Why define tangent vectors as derivations?: The derivation definition is intrinsic and coordinate-free: a tangent vector is a linear operator on smooth functions satisfying the Leibniz rule, which avoids reference to any embedding and works on abstract manifolds.
What does the Lie bracket of two vector fields measure?: It measures the failure of the flows of the two vector fields to commute; vanishing of the bracket means the flows can be followed in either order to reach the same point.