Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

P

P 1

P 2

x

y

f(x, y, α 1 )=0 f(x, y, α 1 +h)=0

Figure 5.4 Two neighbouring curves in thexy-plane of the familyf(x, y, α)=

0 intersecting atP.Forfixedα 1 , the pointP 1 is the limiting position ofPas

h→0. Asα 1 is varied,P 1 delineates the envelope of the family (broken line).

a surface in a curve. Thus different values of the parameterαcorrespond to

different curves, which can be plotted in thexy-plane. We now investigate how

theenvelope equationfor such a family of curves is obtained.

5.10.1 Envelope equations

Supposef(x, y, α 1 )=0andf(x, y, α 1 +h) = 0 are two neighbouring curves of a

family for which the parameterαdiffers by a small amounth. Let them intersect

at the pointPwith coordinatesx, y, as shown in figure 5.4. Then the envelope,

indicated by the broken line in the figure, touchesf(x, y, α 1 ) = 0 at the pointP 1 ,

which is defined as the limiting position ofPwhenα 1 is fixed buth→0. The

full envelope is the curve traced out byP 1 asα 1 changes to generate successive

members of the family of curves. Of course, for any finiteh,f(x, y, α 1 +h)=0is

one of these curves and the envelope touches it at the pointP 2.

We are now going to apply Rolle’s theorem, see subsection 2.1.10, with the

parameterαas the independent variable andxandyfixed as constants. In this

context, the two curves in figure 5.4 can be thought of as the projections onto the

xy-plane of the planar curves in which thesurfacef=f(x, y, α) = 0 meets the

planesα=α 1 andα=α 1 +h.

Along the normal to the page that passes throughP,asαchanges fromα 1

toα 1 +hthe value off=f(x, y, α) will depart from zero, because the normal

meets the surfacef=f(x, y, α)=0onlyatα=α 1 and atα=α 1 +h. However,

at these end points the values off=f(x, y, α) will both be zero, and therefore

equal. This allows us to apply Rolle’s theorem and so to conclude that for some

θin the range 0≤θ≤1 the partial derivative∂f(x, y, α 1 +θh)/∂αis zero. When