Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

From equation (5.5) the total differential off(x, y) is given by

df=

∂f

∂x

dx+

∂f

∂y

dy,

but we now note that by using the formal device of dividing through byduthis

immediately implies

df

du

=

∂f

∂x

dx

du

+

∂f

∂y

dy

du

, (5.14)

which is called thechain rulefor partial differentiation. This expression provides

a direct method for calculating the total derivative offwith respect touand is

particularly useful when an equation is expressed in a parametric form.

Given thatx(u)=1+auandy(u)=bu^3 , find the rate of change off(x, y)=xe−ywith

respect tou.

As discussed above, this problem could be addressed by substituting forxandyto obtain
fas a function only ofuand then differentiating with respect tou. However, using (5.14)
directly we obtain

df

du

=(e−y)a+(−xe−y)3bu^2 ,

which on substituting forxandygives

df

du

=e−bu

3

(a− 3 bu^2 − 3 bau^3 ).

Equation (5.14) is an example of the chain rule for a function of two variables

each of which depends on a single variable. The chain rule may be extended to

functions of many variables, each of which is itself a function of a variableu,i.e.

f(x 1 ,x 2 ,x 3 ,...,xn), withxi=xi(u). In this case the chain rule gives

df

du

=

∑n

i=1

∂f

∂xi

dxi

du

=

∂f

∂x 1

dx 1

du

+

∂f

∂x 2

dx 2

du

+···+

∂f

∂xn

dxn

du

. (5.15)

5.6 Change of variables

It is sometimes necessary or desirable to make a change of variables during the

course of an analysis, and consequently to have to change an equation expressed

in one set of variables into an equation using another set. The same situation arises

if a functionfdepends on one set of variablesxi,sothatf=f(x 1 ,x 2 ,...,xn) but

thexiare themselves functions of a further set of variablesujand given by the

equations

xi=xi(u 1 ,u 2 ,...,um). (5.16)