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"
Mode of a linear field usually the energy of the solution "assuming that there is no other field in S". From the norm are defined the orthogonality and the scalar product of solutions. The orthogonality of solutions extends to the corresponding modes. Mode of a linear field In physics, a vector field is linear if it is a solution of a set of linear equations "E". For instance, in physics, the electromagnetic field in vacuum, defined in the usual (3 + 1)-dimensional space "S", obeys Maxwell's equations. A linear combination of electromagnetic fields, with constant, real coefficients, is a new field which obeys Maxwell's equations.