Swirl and the Curl

field

{-y,x,1}

time

point

{2,0,0}

{2,1,-1}

{0,2,0}

{1.2,1.2,1.}

normal θ

normal ϕ

motion

absolute

relative

motes
paths

curl = {0.,0.,2.}

N = {0.4794,0.,0.8776}

curl · N = 1.75517

Imagine a moving fluid, such as air or water. Each point has a velocity vector, so that all the points make what is called a vector field. At each point in the flow, the curl of the vector field indicates the magnitude and orientation of the swirling of the flow. The curl can be visualized as follows. Near a given point, let there be a given plane containing the point and four nearby points that form a square cross in the plane. As time changes, the five points move out of the plane; let their "shadows" be projected orthogonally onto the plane, and let green lines join opposite points of the projected cross. Finally, let red lines bisecting the angles formed by the green lines be drawn in the plane. As time changes, the green lines move and the red bisectors rotate. The rotation of the red lines indicates the component of the curl in the direction normal to the plane. A succession of three such configurations of green and red lines is drawn on the left and forms a sort of scale; the rotation of the configuration is illustrated as it passes the selected point.