Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

5.8 STATIONARY VALUES OF MANY-VARIABLE FUNCTIONS


P


S


y

x

B


Figure 5.2 Stationary points of a function of two variables. A minimum
occurs atB, a maximum atPand a saddle point atS.

For functions of two variables, such as the one shown, it should be clear that a

necessary condition for a stationary point (maximum, minimum or saddle point)


to occur is that


∂f
∂x

= 0 and

∂f
∂y

=0. (5.21)

The vanishing of the partial derivatives in directions parallel to the axes is enough


to ensure that the partial derivative in any arbitrary direction is also zero. The


latter can be considered as the superposition of two contributions, one along


each axis; since both contributions are zero, so is the partial derivative in the


arbitrary direction. This may be made more precise by considering the total


differential


df=

∂f
∂x

dx+

∂f
∂y

dy.

Using (5.21) we see that although the infinitesimal changesdxanddycan be


chosen independently the change in the value of the infinitesimal functiondfis


always zero at a stationary point.


We now turn our attention to determining the nature of a stationary point of

a function of two variables, i.e. whether it is a maximum, a minimum or a saddle


point. By analogy with the one-variable case we see that∂^2 f/∂x^2 and∂^2 f/∂y^2


must both be positive for a minimum and both be negative for a maximum.


However these are not sufficient conditions since they could also be obeyed at


complicated saddle points. What is important for a minimum (or maximum) is


that the second partial derivative must be positive (or negative) inalldirections,


not just in thex-andy- directions.

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