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
Angular velocity Lie algebra of the Lie group of rotations of an "n"-dimensional inner product space. In three dimensions, angular velocity can be represented by a pseudovector because second rank tensors are dual to pseudovectors in three dimensions. Since the angular velocity tensor "W" = "W"("t") is a skew-symmetric matrix: its Hodge dual is a vector, which is precisely the previous angular velocity vector formula_60. If we know an initial frame "A"(0) and we are given a "constant" angular velocity tensor "W", we can obtain "A"("t") for any given "t". Recall the matrix differential equation: This equation can be integrated to give: