Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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

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