This applet is designed to build intuition about divergence of a vector field.
- The applet has a control window, a Vector Field window, and a NormalFlowAntipodal window.
Vector Field window shows the the vector field V with a viewing window
centered around a point P0 whose coordinates are controlled by the
boxes and buttons of the control panel. A green circle of radius loopscale is
centered at the point. The loopvectors checkbox plots V at points
on this loop. Red arrows from the loop give the normal component
of V, with a red curve drawn at the end of these vectors.
- The control panel lets you define the vector field V=(p,q).
It gives readouts for the value of P0 along with V, and its
divergence, all evaluated at P0.
control panel lets you set the viewing window along with the density
and scaling factor of the vector field. You can also scale the
size of the green circle and the length of the red vectors.
NormalFlowAntipodal window sets up for the integral definition of
divergence. (Recall that if we let C be a circle of radius r
around a point P0, (the divergence at P0)= the limit as r goes to zero of (the integral of the length
of the normal component of the flow)/(Pi*r).) The
NormalFlowAntipodal window looks at the sum of the normal component of
field at antipodal points. As the circle gets smaller we can see
that the projections of
much of the flow from antipodal points cancel out, so that it is
cleaner to look at the integral of the sum of the flows through