Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.10 ENVELOPES

We now have the general form for the distribution of particles amongst energy levels, but
in order to determine the two constantsμ,Cwe recall that

∑R

k=1

CexpμEk=N

and
∑R

k=1

CEkexpμEk=E.

This is known as the Boltzmann distribution and is a well-known result from statistical
mechanics.

5.10 Envelopes

As 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 one

or more variables, quantities that are normally considered as parameters of the

system under study. Such parameters may, for example, represent the capacitance

of a capacitor, the length of a rod, or the mass of a particle – quantities that

are normally taken as fixed for any particular physical set-up. The corresponding

variables may well be time, currents, charges, positions and velocities. However,

the parameterscouldbe 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,

typicallyy=y(x), is affected when the value of a parameter is changed in a

smooth and continuous way. In effect, we are making the parameter into an

additional variable.

As a particular parameter, which we denote byα, is varied over its permitted

range, the shape of the plot ofyagainstxwill change, usually, but not always,

in a smooth and continuous way. For example, if the muzzle speedvof a shell

fired from a gun is increased through a range of values then its height–distance

trajectories will be a series of curves with a common starting point that are

essentially just magnified copies of the original; furthermore the curves do not

cross each other. However, if the muzzle speed is kept constant butθ, the angle

of elevation of the gun, is increased through a series of values, the corresponding

trajectories do not vary in a monotonic way. Whenθhas been increased beyond

45 ◦the trajectories then do cross some of the trajectories corresponding toθ< 45 ◦.

The trajectories forθ> 45 ◦all lie within a curve that touches each individual

trajectory at one point. Such a curve is called theenvelopeto the set of trajectory

solutions; 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 the

formf=f(x, y, α) = 0. A function of three Cartesian variables,f=f(x, y, α),

is defined at all points inxy α-space, whereasf=f(x, y, α)=0isasurfacein

this space. A plane of constantα, which is parallel to thexy-plane, cuts such