Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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)
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