Mathematical Tools for Physics

8—Multivariable Calculus 227

8.11 Maxima, Minima, Saddles
With one variable you can look for a maximum or a minimum by taking a derivative and setting it to zero. For
several variables you do it several times so that you will get as many equations as you have unknown coordinates.
Put this in the language of gradients:∇f= 0. The derivative offvanishes in every direction as you move
from such a point. As examples,

f(x,y) =x^2 +y^2 , or =−x^2 −y^2 , or =x^2 −y^2

For all three of these the gradient is zero at(x,y) = (0,0); the first has a minimum there, the second a maximum,
and the third neither — it is a “saddle point.” Draw a picture to see the reason for the name. The generic term
for all three of these is “critical point.”
An important example of finding a minimum is “least square fitting” of functions. How close are two
functions to each other? The most commonly used, and in every way the simplest, definition of the distance
betweenfandgon the intervala < x < bis

∫b

a

dx

∣

∣f(x)−g(x)

∣

∣^2 (20)

This means that a large deviation of one function from the other in a small region counts more than smaller
deviations spread over a larger domain. The square sees to that. As a specific example, I have a functionfon
the interval 0 < x < Land I want to fit it to the sum of a couple of trigonometric functions. The best fit will
be the one that minimizes the distance betweenfand the sum. (Takefto be a real-valued function for now.)

D(α,β) =

∫L

0

dx

(

f(x)−αsin

πx

L

−βsin

2 πx

L

) 2

(21)

Dis the distance between the given function and the sines that I want to fit to it. To minimize the distance,
take derivatives with respect to the parametersαandβ.

dD

dα

= 2

∫L

0

dx

(

f(x)−αsin

πx

L

−βsin

2 πx

L

)(

−sin

πx

L

)

= 0

dD

dβ

= 2

∫L

0

dx

(

f(x)−αsin

πx

L

−βsin

2 πx

L

)(

−sin

2 πx

L

)

= 0