Variable-mass system events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass "m" moves at a velocity v at an initial time "t". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time "t" + d"t", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d"m"dv is the product of two small values, it can be ignored, meaning during d"t"
Cross product The cross product with n therefore describes the infinitesimal generator of the rotations about n. These infinitesimal generators form the Lie algebra so(3) of the rotation group SO(3), and we obtain the result that the Lie algebra R with cross product is isomorphic to the Lie algebra so(3). The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector: where superscript refers to the transpose operation, and [a] is defined by: The columns [a] of the skew-symmetric matrix for a vector a can be also obtained by calculating the cross product with unit