The white vectors shown in this demo are the normal vectors given by (-g_x(x, y), -g_y(x, y), 1) for the surface g(x, y). The red vectors are the part of the vector field F(x, y, z) that lies on this surface. The integral of the vector field F(x, y, z) over this surface for some domain D is equivalent to the integral of the dot product of the white and red vectors times dxdy.