Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

5.9 STATIONARY VALUES UNDER CONSTRAINTS


where thearare coefficients dependent upon ∆x. Substituting this into (5.26), we


find


∆f=^12 ∆xTM∆x=^12


r

λra^2 r.

Now, for the stationary point to be a minimum, we require ∆f=^12



rλra

2
r>^0
for all sets of values of thear, and therefore all the eigenvalues ofMto be


greater than zero. Conversely, for a maximum we require ∆f=^12



rλra

2
r<0,
and therefore all the eigenvalues ofMto be less than zero. If the eigenvalues have


mixed signs, then we have a saddle point. Note that the test may fail if some or


all of the eigenvalues are equal to zero and all the non-zero ones have the same


sign.


Derive the conditions for maxima, minima and saddle points for a function of two real
variables, using the above analysis.

For a two-variable function the matrixMis given by


M=

(


fxx fxy
fyx fyy

)


.


Therefore its eigenvalues satisfy the equation
∣∣


fxx−λfxy
fxy fyy−λ

∣∣



∣=0.


Hence


(fxx−λ)(fyy−λ)−f^2 xy=0

⇒ fxxfyy−(fxx+fyy)λ+λ^2 −f^2 xy=0

⇒ 2 λ=(fxx+fyy)±


(fxx+fyy)^2 −4(fxxfyy−fxy^2 ),

which by rearrangement of the terms under the square root gives


2 λ=(fxx+fyy)±


(fxx−fyy)^2 +4f^2 xy.

Now, thatMis real and symmetric implies that its eigenvalues are real, and so for both
eigenvalues to be positive (corresponding to a minimum), we requirefxxandfyypositive
and also


fxx+fyy>


(fxx+fyy)^2 −4(fxxfyy−fxy^2 ),

⇒ fxxfyy−fxy^2 > 0.

A similar procedure will find the criteria for maxima and saddle points.


5.9 Stationary values under constraints

In the previous section we looked at the problem of finding stationary values of


a function of two or more variables when all the variables may be independently

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