Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

theorem then becomes

f(x)=f(x 0 )+

∑

i

∂f

∂xi

∆xi+

1

2!

∑

i

∑

j

∂^2 f

∂xi∂xj

∆xi∆xj+···,

(5.20)

where ∆xi=xi−xi 0 and the partial derivatives are evaluated at (x 10 ,x 20 ,...,xn 0 ).

For completeness, we note that in this case the full Taylor series can be written

in the form

f(x)=

∑∞

n=0

1

n!

[

(∆x·∇)nf(x)

]

x=x 0 ,

where∇is the vector differential operator del, to be discussed in chapter 10.

5.8 Stationary values of many-variable functions

The idea of thestationary pointsof a function of just one variable has already

been discussed in subsection 2.1.8. We recall that the functionf(x) has a stationary

point atx=x 0 if its gradientdf/dxis zero at that point. A function may have

any number of stationary points, and their nature, i.e. whether they are maxima,

minima or stationary points of inflection, is determined by the value of the second

derivative at the point. A stationary point is

(i) a minimum ifd^2 f/dx^2 >0;

(ii) a maximum ifd^2 f/dx^2 <0;

(iii) a stationary point of inflection ifd^2 f/dx^2 = 0 and changes sign through

the point.

We now consider the stationary points of functions of more than one variable;

we will see that partial differential analysis is ideally suited to the determination

of the position and nature of such points. It is helpful to consider first the case

of a function of just two variables but, even in this case, the general situation

is more complex than that for a function of one variable, as can be seen from

figure 5.2.

This figure shows part of a three-dimensional model of a functionf(x, y). At

positionsPandBthere are a peak and a bowl respectively or, more mathemati-

cally, a local maximum and a local minimum. At positionSthe gradient in any

direction is zero but the situation is complicated, since a section parallel to the

planex= 0 would show a maximum, but one parallel to the planey=0would

show a minimum. A point such asSis known as asaddle point. The orientation

of the ‘saddle’ in thexy-plane is irrelevant; it is as shown in the figure solely for

ease of discussion. For any saddle point the function increases in some directions

away from the point but decreases in other directions.