vector field

Look at a weather map dotted with little arrows: at every spot the arrow shows which way the wind blows and how hard. That picture — an arrow attached to every point in space — is a vector field. Unlike an ordinary function that returns a single number at each point, a vector field returns a whole arrow, carrying both a direction and a strength. Flowing water, magnetic forces, gravitational pull, and air currents are all naturally described this way.

Formally, a vector field assigns to each point a vector. In two dimensions you write F(x, y) = ( P(x, y), Q(x, y) ), where the two component functions give the arrow's horizontal and vertical parts at that point; in three dimensions there is a third component. So a vector field is really several ordinary functions packaged together, drawn as a field of arrows whose lengths and directions vary smoothly from place to place.

Vector fields are the stage on which vector calculus is performed. You can measure how much a field pushes along a path (a line integral, which gives the work done by a force), and you can take two special derivatives of a field — its divergence and its curl — to ask whether the field is spreading apart or swirling around. These ideas underpin fluid dynamics and the whole of electromagnetism.