Vector field on a Euclidean open set

A vector field on an open set $U\subseteq\mathbb R^n$ is a map $u:U\to\mathbb R^n$ . In Cartesian coordinates it is written $u=(u_1,\ldots,u_n)$ . The word “field” alone imposes no regularity: continuity, differentiability, measurability or integrability must be specified.

Time and regularity

A time-dependent field is a map $(t,x)\mapsto u(t,x)$ . A smooth field has smooth Cartesian components. In a moving orthonormal basis, its scalar components can satisfy different derivative formulas because the basis itself varies.

Geometric interpretation

Identifying every tangent space of $U$ with $\mathbb R^n$ identifies a smooth Euclidean field with a smooth tangent vector field. The elementary definition here does not require a choice of manifold charts.