Areal velocity equal to the magnitude of the cross product of vectors "AB" and "AC". This area can also be viewed as a vector with this magnitude, pointing in a direction perpendicular to the parallelogram; this vector is the cross product itself: Hence The areal velocity is this vector area divided by Δ"t" in the limit that Δ"t" becomes vanishingly small: But, formula_4 is the velocity vector formula_5 of the moving particle, so that On the other hand, the angular momentum of the particle is and hence the angular momentum equals 2"m" times the areal velocity. Conservation of areal velocity is a
Mass flow rate product is as follows. The only mass flowing "through" the cross-section is the amount normal to the area, i.e. parallel to the unit normal. This amount is: where "θ" is the angle between the unit normal formula_4 and the velocity of mass elements. The amount passing through the cross-section is reduced by the factor formula_5, as "θ" increases less mass passes through. All mass which passes in tangential directions to the area, that is perpendicular to the unit normal, "doesn't" actually pass "through" the area, so the mass passing through the area is zero. This occurs when "θ" = "π"/2: