Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

toxandyrespectively, and they are extremely important in a wide range of

physical applications.

For a function of two variablesf(x, y) we may define the derivative with respect

tox, for example, by saying that it is that for a one-variable function whenyis

held fixed and treated as a constant. To signify that a derivative is with respect

tox, but at the same time to recognize that a derivative with respect toyalso

exists, the former is denoted by∂f/∂xand is thepartial derivative off(x, y)with

respect tox. Similarly, the partial derivative offwith respect toyis denoted by

∂f/∂y.

To define formally the partial derivative off(x, y) with respect tox, we have

∂f

∂x

= lim

∆x→ 0

f(x+∆x, y)−f(x, y)

∆x

, (5.1)

provided that the limit exists. This is much the same as for the derivative of a

one-variable function. The other partial derivative off(x, y) is similarly defined

as a limit (provided it exists):

∂f

∂y

= lim

∆y→ 0

f(x, y+∆y)−f(x, y)

∆y

. (5.2)

It is common practice in connection with partial derivatives of functions

involving more than one variable to indicate those variables that are held constant

by writing them as subscripts to the derivative symbol. Thus, the partial derivatives

defined in (5.1) and (5.2) would be written respectively as
(
∂f
∂x

)

y

and

(

∂f

∂y

)

x

.

In this form, the subscript shows explicitly which variable is to be kept constant.

A more compact notation for these partial derivatives isfxandfy. However, it is

extremely important when using partial derivatives to remember which variables

are being held constant and it is wise to write out the partial derivative in explicit

form if there is any possibility of confusion.

The extension of the definitions (5.1), (5.2) to the generaln-variable case is

straightforward and can be written formally as

∂f(x 1 ,x 2 ,...,xn)

∂xi

= lim

∆xi→ 0

[f(x 1 ,x 2 ,...,xi+∆xi,...,xn)−f(x 1 ,x 2 ,...,xi,...,xn)]

∆xi

,

provided that the limit exists.

Just as for one-variable functions, second (and higher) partial derivatives may

be defined in a similar way. For a two-variable functionf(x, y)theyare

∂

∂x

(

∂f

∂x

)

=

∂^2 f

∂x^2

=fxx,

∂

∂y

(

∂f

∂y

)

=

∂^2 f

∂y^2

=fyy,

∂

∂x

(

∂f

∂y

)

=

∂^2 f

∂x∂y

=fxy,

∂

∂y

(

∂f

∂x

)

=

∂^2 f

∂y∂x

=fyx.