Finally, flux as a vector field : j ( p ) = ∂ ∂ I ∂ ∂ A ( p ) , I ( A , p ) = a r g m a x n ^ ^ n ^ ^ p d q d t ( A , p , n ^ ^ ) . In this case, there is no fixed surface we are measuring over. q is a function of a point, an area, and a direction (given by a unit vector n ^ ^ ), and measures the flow through the disk of area A perpendicular to that unit vector. I is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an abuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.)