vector field

A function that assigns an arrow — a vector with its own length and direction — to every point in a region, instead of a single number.

A vector field takes a point $(x,y)$ and returns a vector $\mathbf{F}(x,y) = P(x,y)\,\mathbf{i} + Q(x,y)\,\mathbf{j}$ instead of a single number — an arrow with its own length and direction, drawn with its tail sitting at the point that produced it. $\mathbf{F}$ is really just a pair of ordinary scalar functions $P$ and $Q$ bolted together: $P$ says how far to go in $x$ , $Q$ says how far to go in $y$ , both evaluated at the same input point. A field's length $|\mathbf{F}(x,y)| = \sqrt{P^2+Q^2}$ carries real meaning (wind speed, force magnitude), distinct from its direction.

Examples

Wind speed and direction over a map, the pull of gravity around a planet, and the current swirling in a river bend are all vector fields.
$\mathbf{F}(x,y)=(-y,x)$ is a rotational field where every arrow is perpendicular to the position vector.