Mathematical Methods for Physics and Engineering - Matematica.NET

5.10 ENVELOPESWe now have the general form for the distribution of particles amongst energy levels, butin order to determine the two constants µ, C we recall thatandR∑C exp µE k = Nk=1R∑CE k exp µE k = E.k=1This is known as the Boltzmann distribution and is a well-known result from statisticalmechanics. ◭5.10 EnvelopesAs noted at the start of this chapter, many of the functions with which physicists,chemists and engineers have to deal contain, in addition to constants and oneor more variables, quantities that are normally considered as parameters of thesystem under study. Such parameters may, for example, represent the capacitanceof a capacitor, the length of a rod, or the mass of a particle – quantities thatare normally taken as fixed for any particular physical set-up. The correspondingvariables may well be time, currents, charges, positions and velocities. However,the parameters could be varied and in this section we study the effects of doing so;in particular we study how the form of dependence of one variable on another,typically y = y(x), is affected when the value of a parameter is changed in asmooth and continuous way. In effect, we are making the parameter into anadditional variable.As a particular parameter, which we denote by α, is varied over its permittedrange, the shape of the plot of y against x will change, usually, but not always,in a smooth and continuous way. For example, if the muzzle speed v of a shellfired from a gun is increased through a range of values then its height–distancetrajectories will be a series of curves with a common starting point that areessentially just magnified copies of the original; furthermore the curves do notcross each other. However, if the muzzle speed is kept constant but θ, the angleof elevation of the gun, is increased through a series of values, the correspondingtrajectories do not vary in a monotonic way. When θ has been increased beyond45 ◦ the trajectories then do cross some of the trajectories corresponding to θ45 ◦ all lie within a curve that touches each individualtrajectory at one point. Such a curve is called the envelope to the set of trajectorysolutions; it is to the study of such envelopes that this section is devoted.For our general discussion of envelopes we will consider an equation of theform f = f(x, y, α) = 0. A function of three Cartesian variables, f = f(x, y, α),is defined at all points in xyα-space, whereas f = f(x, y, α) =0isasurface inthis space. A plane of constant α, which is parallel to the xy-plane, cuts such173