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"
Proof of the Euler product formula for the Riemann zeta function arithmetic, the partial product when expanded out gives a sum consisting of those terms "n" where "n" is a product of primes less than or equal to "q". The inequality results from the fact that therefore only integers larger than "q" can fail to appear in this expanded out partial product. Since the difference between the partial product and ζ("s") goes to zero when σ > 1, we have convergence in this region. Proof of the Euler product formula for the Riemann zeta function Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis "Variae