James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 15.9 Change of Variables in Multiple Integrals 1053
A change of variables can also be useful in double integrals. We have already seen one
example of this: conversion to polar coordinates. The new variables r and are related to
the old variables x and y by the equations
x − r cos
y − r sin
and the change of variables formula (15.3.2) can be written as
y
R
y f sx, yd dA − y
S
y f sr cos , r sin d r dr d
where S is the region in the r-plane that corresponds to the region R in the xy-plane.
More generally, we consider a change of variables that is given by a transformation
T from the uv-plane to the xy-plane:
Tsu, vd − sx, yd
where x and y are related to u and v by the equations
3 x − tsu, vd y − hsu, vd
or, as we sometimes write,
x − xsu, vd
y − ysu, vd
We usually assume that T is a C 1 transformation, which means that t and h have continuous
first-order partial derivatives.
A transformation T is really just a function whose domain and range are both subsets
of R 2 . If Tsu 1 , v 1 d − sx 1 , y 1 d, then the point sx 1 , y 1 d is called the image of the point
su 1 , v 1 d. If no two points have the same image, T is called one-to-one. Figure 1 shows the
effect of a transformation T on a region S in the uv-plane. T transforms S into a region R
in the xy-plane called the image of S, consisting of the images of all points in S.
√
y
S
(u¡, √¡)
T
T–!
R
(x¡, y¡)
FIGURE 1
0
u
0
x
If T is a one-to-one transformation, then it has an inverse transformation T 21 from
the xy-plane to the uv-plane and it may be possible to solve Equations 3 for u and v in
terms of x and y:
u − Gsx, yd v − Hsx, yd
ExamplE 1 A transformation is defined by the equations
x − u 2 2 v 2
y − 2uv
Find the image of the square S − hsu, vd | 0 < u < 1, 0 < v < 1j.
SOLUtion The transformation maps the boundary of S into the boundary of the image.
So we begin by finding the images of the sides of S. The first side, S 1 , is given by v − 0
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