A Visual Divergence Applet

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.
• The 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.
• The 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.
• The 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 the 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 antipodal points.

Things to notice:
• The field (x,y) has a steady divergence of 2.
• The field (x^2,y) has constant divergence on vertical lines
• The field (-y,x) has a steady divergence of 0.  The flow in at one point is canceled by the flow out at the antipodal point.
• The field (2,3) has steady divergence of 0.
• The divergence is always positive at a source and negative at a sink, but divergence need not reach a minimum or maximum at sources and sinks.
• The field (x+.2-y, y+.3+x^2) is a nice field to work with.  It is obviously the sum of a three fields, each of which has a different kind of behavior.  It is a complicate enough field that avoid obvious bad conjectures.

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.