In this demo, we show the flux across a surfaceS(t,s). The window "The unit normals" shows the vecor field V(x,y,z) and the surface S; the red vectors are the unit normals at certain points on the surface. The second window, "V(x,y) at C(t)" shows the vector field V(x,y) evaluated at (x,y,z)=S(t,s).

In the third window we see the graph of the function that gives the value of the dot product of V(S(t,s)) with the unit normal at S(t,s). The flux across S is then equal to the (signed) volume under the curve, and you can read its value off the readout "Flux".