Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

5.6 CHANGE OF VARIABLES


x

y

ρ
φ

Figure 5.1 The relationship betweenCartesian and planepolar coordinates.

For each different value ofi,xiwill be a different function of theuj.Inthiscase


the chain rule (5.15) becomes


∂f
∂uj

=

∑n

i=1

∂f
∂xi

∂xi
∂uj

,j=1, 2 ,...,m, (5.17)

and is said to express achange of variables. In general the number of variables


in each set need not be equal, i.e.mneed not equaln, but if both thexiand the


uiare sets of independent variables thenm=n.


Plane polar coordinates,ρandφ, and Cartesian coordinates,xandy,arerelatedbythe
expressions
x=ρcosφ, y=ρsinφ,
as can be seen from figure 5.1. An arbitrary functionf(x, y)canbere-expressedasa
functiong(ρ, φ). Transform the expression
∂^2 f
∂x^2

+


∂^2 f
∂y^2
into one inρandφ.

We first note thatρ^2 =x^2 +y^2 ,φ=tan−^1 (y/x). We can now write down the four partial
derivatives


∂ρ
∂x

=


x
(x^2 +y^2 )^1 /^2

=cosφ,

∂φ
∂x

=


−(y/x^2 )
1+(y/x)^2

=−


sinφ
ρ

,


∂ρ
∂y

=


y
(x^2 +y^2 )^1 /^2

=sinφ,

∂φ
∂y

=


1 /x
1+(y/x)^2

=


cosφ
ρ

.

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