Flux is defined as the dot product of the vector field F with the Area vector of the surface, as FAcos(theta). You can intuitively visualize this as the vector field flow across the planar surface/fish screen. If theta == 0, you maximize the flow through the fish screen. If theta == 90, you minimize the flow through the fish screen so the flux is zero.
Now to generalize this across 3D surfaces, we take the surface integral of the flux, which is the double integral of the dot product of F with the normal vector across the parametrized 3D surface.
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