Surface integral of a vector field (flux)
We can parametrize a surface over a rectangular domain :
.
Suppose that is a vector field in three-dimensional space
defined at every point on surface . Let the unit normal vectors normal to surface . The surface integral of over the surface can be calculated
.
This integral is called the flux of the vector field across .
In this interactive figure, move the smaller green point throughout the -region . You will see the value of at this point.
Move the larger blue point from bottom left to top right to "integrate." When the blue point is in the top-right corner, the value of the surface integral is shown (this is a numerical approximation).