Change of variables

PARTIAL DIFFERENTIATION
From equation (5.5) the total differential of f(x, y) is given by
df =
∂f
∂f
dx +
dy,
∂x
∂y
but we now note that by using the formal device of dividing through by du this
immediately implies
∂f dx ∂f dy
df
=
+
,
du
∂x du ∂y du
(5.14)
which is called the chain rule for partial differentiation. This expression provides
a direct method for calculating the total derivative of f with respect to u and is
particularly useful when an equation is expressed in a parametric form.
Given that x(u) = 1 + au and y(u) = bu3 , find the rate of change of f(x, y) = xe−y with
respect to u.
As discussed above, this problem could be addressed by substituting for x and y to obtain
f as a function only of u and then differentiating with respect to u. However, using (5.14)
directly we obtain
df
= (e−y )a + (−xe−y )3bu2 ,
du
which on substituting for x and y gives
df
3
= e−bu (a − 3bu2 − 3bau3 ). du
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 variable u, i.e.
f(x1 , x2 , x3 , . . . , xn ), with xi = xi (u). In this case the chain rule gives
∂f dxi
df
∂f dx1
∂f dx2
∂f dxn
=
=
+
+ ···+
.
du
∂xi du
∂x1 du
∂x2 du
∂xn du
n
(5.15)
i=1
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 function f depends on one set of variables xi , so that f = f(x1 , x2 , . . . , xn ) but
the xi are themselves functions of a further set of variables uj and given by the
equations
xi = xi (u1 , u2 , . . . , um ).
158
(5.16)
5.6 CHANGE OF VARIABLES
y
ρ
φ
x
Figure 5.1 The relationship between Cartesian and plane polar coordinates.
For each different value of i, xi will be a different function of the uj . In this case
the chain rule (5.15) becomes
∂f ∂xi
∂f
=
,
∂uj
∂xi ∂uj
n
j = 1, 2, . . . , m,
(5.17)
i=1
and is said to express a change of variables. In general the number of variables
in each set need not be equal, i.e. m need not equal n, but if both the xi and the
ui are sets of independent variables then m = n.
Plane polar coordinates, ρ and φ, and Cartesian coordinates, x and y, are related by the
expressions
x = ρ cos φ,
y = ρ sin φ,
as can be seen from figure 5.1. An arbitrary function f(x, y) can be re-expressed as a
function g(ρ, φ). Transform the expression
∂2 f
∂2 f
+ 2
2
∂x
∂y
into one in ρ and φ.
We first note that ρ2 = x2 + y 2 , φ = tan−1 (y/x). We can now write down the four partial
derivatives
∂ρ
x
= cos φ,
= 2
∂x
(x + y 2 )1/2
∂φ
sin φ
−(y/x2 )
=−
=
,
∂x
1 + (y/x)2
ρ
y
∂ρ
= sin φ,
= 2
∂y
(x + y 2 )1/2
∂φ
cos φ
1/x
=
=
.
∂y
1 + (y/x)2
ρ
159