Vector field

Definition

Let $S \subseteq \mathbb{R}^n$. A vector field on $S$ is a vector-valued function $F \colon S \to \mathbb{R}^n$, $F(x) = (F_1(x), \dots, F_n(x))$. The vector field is continuous, or of class $C^k$, if each component $F_i$ is continuous, or of class C^k, on $S$.

Remarks

One pictures $F$ by an arrow $F(x)$ drawn with its tail at the point $x$. For a differentiable function $f \colon U \to \mathbb{R}$ on an open set $U$, the gradient $\nabla f$ is a vector field on $U$. Such fields are gradient fields. The field $F(x, y) = (-y, x)$ on $\mathbb{R}^2$ describes a counterclockwise rotation about the origin. The field $F(x) = x$ points away from the origin. In $\mathbb{R}^3$ one often writes $F = P \, \mathbf{i} + Q \, \mathbf{j} + R \, \mathbf{k}$, with $P$, $Q$, $R$ the components of $F$.

Explanations

L1 human

A vector field attaches an arrow to each point. Think of the wind on a weather map: at each place there is a wind direction and a wind speed.