288 Chapter 9Functions of several variables
The general case
The transformation from cartesian to polar coordinates is a special case of the general
transformation of variables (of coordinate system). It is also the most important in
practice, and only a brief discussion of the general case is given here. Let
x 1 = 1 x(u, 1 v), y 1 = 1 y(u, 1 v) (9.61)
be continuous and differentiable functions of the variables uand v, such that for each
point(x,y)in a region R of the xy-plane there corresponds a unique point(u, 1 v)in
the corresponding region R*of the uv-plane. Then
(9.62)
where|J|is the modulus of the Jacobianof the transformation,
10(9.63)
For example, ifx 1 = 1 r 1 cos 1 θandy 1 = 1 r 1 sin 1 θthen
anddA 1 = 1 dxdy 1 → 1 dA 1 = 1 rdrdθ, as obtained geometrically from Figure 9.16. The Jacobian
is used in advanced formulations of thermodynamics for the transformation of
thermodynamic variables. The method can be generalized for three or more variables.
The integral
It was stated in Example 6.16 of Section 6.5 that this integral cannot be evaluated by
the methods described in Chapters 5 and 6. It can however by evaluated by means of
a ‘trick’ involving the transformation of a double integral from cartesian to polar
coordinates.
11We have
because the integral is an even function of x(see Section 5.3). Also
Iedy
y=
−−1
2
2Z
∞∞Iedx edx
xx==
−−−ZZ
022
12
∞∞∞Z
02∞edx
−xJ
x
r
yxy
r
= rrr
∂
∂
∂
∂
−
∂
∂
∂
∂
=+=
θθ
cos θθsin
22J
xy
u
x
u
yxy
u
=
∂,
∂,
=
∂
∂
∂
∂
−
∂
∂
∂
∂
()
()vvv
ZZ ZZ
RR*f x ydxdy(),=f xu yu Jdud()()(),,,vv v
10Carl Gustave Jacobi (1804–1851), born in Berlin, discussed the quantities called Jacobians in his De
determinantibus functionalibus(On functional determinants) of 1841. The general transformation of variables in a
double integral was given by Euler in 1769.
11The integral was first evaluated by de Moivre in 1733. Laplace, following Euler, used the method described
here in 1774.