Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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

.