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
Velocity gradient analysis, the dimensions of velocity gradient can be determined. The dimensions of velocity are formula_4 , and the dimensions of distance are formula_5. Since the velocity gradient can be expressed as formula_6. Therefore, the velocity gradient has the same dimensions as this ratio, i.e. formula_7. In 3 dimensions, the gradient, formula_8 , of the velocity formula_9 is a second-order tensor which can be expressed as the matrix L: formula_11 can be decomposed into the sum of a symmetric matrix formula_12 and a skew-symmetric matrix formula_13 as follows formula_12 is called the strain rate tensor and describes the rate of stretching