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
first graphical window is the domain window which shows a grid in s and
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.
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.