Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

PARTIAL DIFFERENTIATION


varied. However, it is often the case in physical problems that not all the vari-


ables used to describe a situation are in fact independent, i.e. some relationship


between the variables must be satisfied. For example, if we walk through a hilly


landscape and we are constrained to walk along a path, we will never reach


the highest peak on the landscape unless the path happens to take us to it.


Nevertheless, we can still find the highest point that we have reached during our


journey.


We first discuss the case of a function of just two variables. Let us consider

finding the maximum value of the differentiable functionf(x, y) subject to the


constraintg(x, y)=c,wherecis a constant. In the above analogy,f(x, y) might


represent the height of the land above sea-level in some hilly region, whilst


g(x, y)=cis the equation of the path along which we walk.


We could, of course, use the constraintg(x, y)=cto substitute forxoryin

f(x, y), thereby obtaining a new function of only one variable whose stationary


points could be found using the methods discussed in subsection 2.1.8. However,


such a procedure can involve a lot of algebra and becomes very tedious for func-


tions of more than two variables. A more direct method for solving such problems


is themethod of Lagrange undetermined multipliers, which we now discuss.


To maximisefwe require

df=

∂f
∂x

dx+

∂f
∂y

dy=0.

Ifdxanddywere independent, we could concludefx=0=fy. However, here


they are not independent, but constrained becausegis constant:


dg=

∂g
∂x

dx+

∂g
∂y

dy=0.

Multiplyingdgby an as yet unknown numberλand adding it todfwe obtain


d(f+λg)=

(
∂f
∂x


∂g
∂x

)
dx+

(
∂f
∂y


∂g
∂y

)
dy=0,

whereλis called aLagrange undetermined multiplier. In this equationdxanddy


are to be independent and arbitrary; we must therefore chooseλsuch that


∂f
∂x


∂g
∂x

=0, (5.27)

∂f
∂y


∂g
∂y

=0. (5.28)

These equations, together with the constraintg(x, y)=c, are sufficient to find the


three unknowns, i.e.λand the values ofxandyat the stationary point.

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