A Flux Integral Applet

This applet is designed to build intuition about flux integrals.  We are looking at the integral , where F is a vector field and R is a region parameterized by s and t.  To evaluate the integral we want to convert it to a simple double integral of a real valued function over a region in s and t.
• The applet has a control window and 3 graphics windows.
• The first graphical window is the domain window which shows a grid in s and t.  It contains a movable red hot spot P that shows up in all three graphs as well blue and orange lines through P with s and t respectively kept constant, that also show up in the other graphs.
• The second graphical window is the surface window.  It transforms the domain grid into a parameterized surface.  The red dot P0(P) is the image of P.  There are also tangent vectors to the two curves through P with s and t held constant.  The cross product of the tangent vectors is the red normal vector.  The field F evaluated at P0(P) is a purple vector, and the  light blue line in the normal direction has length equal to the integrand,  or of the dot product of he red and purple vectors.
• The third window is the integrand window.  It reduces the flux integral to the double integral of a real valued function over a region in s and t.  The graph of that function is given.
• The control window lets you specify the function F and the parameterized surface.  Check boxes let you add in the vectors for the vector field, or for the field and the projection onto the normal vector for either the surface or for red and orange curves.  Readouts give Pst, the s and t coordinates of P; P0P, the coordinates of the point on the surface to which P maps; FP, the value of F at P0P; NormalP, the value of the normal vector at P0P, IntegrandP, the dot product of NormalP and FP; and Integral, the value of the flux interval.

Things to notice:
• The default vector field is F = (x, z, y) and the default surface is an eighth of the unit sphere.
• For vector fields like F = (z-y,x-z, y-x), that are the curls of a function, the integral is zero for any simply closed surface of integration.

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.