7.1: Velocity Fields

Suppose we have water flowing in a region $A$ of the plane. Then at every point $(x, y)$ in $A$ the water has a velocity. In general, this velocity will change with time. We’ll let $F$ stand for the velocity vector field and we can write

$$F(x, y, t) = (u(x, y, t), v(x, y, t)). \nonumber$$

The arguments $(x, y, t)$ indicate that the velocity depends on these three variables. In general, we will shorten the name to velocity field (Figure $\PageIndex{1}$).

Figure $\PageIndex{1}$: Streamlines of wind direction over North America 2 February 2009. (CC BY-SA 3.0; Cloudruns via Wikipedia)

A dynamic beautiful and mesmerizing example of a velocity field is at http://hint.fm/wind/index.html. This shows the current velocity of the wind at all points in the continental U.S.