Vector Fields

Core Idea

A vector field on a smooth manifold M is a smooth assignment of a tangent vector to each point — formally, a smooth section of the tangent bundle. In local coordinates, a vector field looks like X = Xⁱ(x)∂/∂xⁱ where the coefficient functions Xⁱ are smooth. Vector fields act as derivations on the algebra of smooth functions, generating flows (one-parameter families of diffeomorphisms) that describe how points move along the field.

Explainer

A tangent vector lives at a single point. A vector field is the global version: a rule that assigns a tangent vector X_p ∈ TpM to every point p ∈ M, smoothly. In local coordinates (x¹, ..., xⁿ), this means X = Xⁱ(x) ∂/∂xⁱ where the component functions Xⁱ are smooth real-valued functions. You can think of a vector field as an arrow attached to every point of the manifold, varying smoothly from point to point — like a wind map or a velocity field in fluid dynamics.

Vector fields act on smooth functions: given f ∈ C∞(M), the function Xf defined by (Xf)(p) = X_p(f) is again smooth. This makes each vector field a derivation of the algebra C∞(M) — a linear map satisfying the Leibniz rule X(fg) = f·X(g) + g·X(f). In fact, derivations of C∞(M) are in bijection with smooth vector fields, so you can equivalently define a vector field as a derivation of the function algebra. This algebraic perspective becomes essential when defining Lie brackets and connections.

The flow of a vector field is the family of diffeomorphisms φt : M → M obtained by following integral curves for time t. An integral curve γ(t) satisfies γ'(t) = X_γ(t) — at each moment, its velocity equals the vector field at its current position. The existence and uniqueness theorem for ODEs guarantees that through each point there passes a unique integral curve, at least for short time. On compact manifolds, the flow exists for all time (the vector field is complete). The flow satisfies the group property φs ∘ φt = φs+t, making it a one-parameter group of diffeomorphisms.

The space of all smooth vector fields on M, denoted 𝔛(M) or Γ(TM), has a rich algebraic structure. It is an infinite-dimensional real vector space, and more importantly, a module over C∞(M) — you can multiply vector fields by smooth functions. This module structure is what distinguishes tensor operations from non-tensor operations: a map on vector fields is tensorial (defines a tensor) if and only if it is C∞(M)-linear. The Lie bracket, which measures the failure of flows to commute, is a fundamental non-tensorial operation that we encounter next.