Variable-mass system the momentum of the system varies for Therefore, by Newton's second law Noting that u - v is the velocity of d"m" relative to "m", symbolized as v, this final equation can be arranged as In a system where mass is being ejected or ablated from a main body, the derivation is slightly different. At time "t", let a mass "m" travel at a velocity v, meaning the initial momentum of the system is Since the main body will be losing mass, d"m" will be negative, meaning that at a time "t" + d"t" the momentum of the system becomes
Tensor product of modules in its right argument: In the general case, each element of the tensor product of modules gives rise to a left "R"-linear map, to a right "R"-linear map, and to an "R"-bilinear form. Unlike the commutative case, in the general case the tensor product is not an "R"-module, and thus does not support scalar multiplication. Both cases hold for general modules, and become isomorphisms if the modules "E" and "F" are restricted to being finitely generated projective modules (in particular free modules of finite ranks). Thus, an element of a tensor product of modules over a ring "R" maps canonically