Lecture 18  Monday May 11th
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Key words: Change of variables, Jacobian, polar, cylindrical and spherical
coordinates
Key concepts: Know how to change variables in integrals
18.1 Change of variables
In this section we give the change of variables theorem for multiple integrals. This is the
analog of the
substitution rule
for integrals of functions of one variable. Suppose that we
want to make the substitution
x
=
x
(
u
) in the integral of a function
f
from
a
to
b
, and
that for
a
≤
x
≤
b
, we have
c
≤
u
≤
d
. Then
Z
b
a
f
(
x
)
dx
=
Z
d
c
f
(
x
(
u
))
dx
du
du.
The important term to remember is the
dx/du
term, and this term has an analog for
multiple integrals.
The function
x
=
x
(
u
) given above is actually a
bijection
from [
c, d
] to [
a, b
]: this means
that for every point
t
∈
[
a, b
], there is a unique number
u
∈
[
c, d
] such that
x
(
u
) =
t
.
That
x
(
u
) is a bijection is essential in order to make the change of integration, and it
means that as a function,
x
(
u
) has an inverse, denoted
x

1
(
u
).
The inverse is itself a
bijection from [
a, b
] to [
c, d
]. For example, the function
x
(
u
) =
u
3
is clearly a bijection
from
R
to
R
, since every real number has a unique real cuberoot. The inverse is given
by
x

1
(
u
) =
u
1
/
3
–
x
(
x

1
(
u
)) =
u
in this case. The function
x
(
u
) =
u
2
is not a bijection,
since
x
(1) =
x
(

1) = 1. To say that a function
x
(
u
) :
R
→
R
is a bijection is equivalent
to saying that the curve representing
x
(
u
) in the
ux
plane passes both the horizontal and
vertical line tests.
For change of variables in multiple integrals, we have a function
f
(
x
) :
R
n
→
R
which we
are integrating over a region
D
, and we wish to replace
x
with a new vector of variables
u
∈
R
n
via the change of variables
x
=
x
(
u
). We suppose that
E
is the domain of
x
(
u
)
and that
x
(
u
) as a function from
E
to
D
is a bijection.
The
inverse function theorem
states that if we want to see whether
x
(
u
) is a bijection, then we check that det
∇
x
(
u
)
6
= 0
for all
u
∈
E
. If that is the case, then
E
=
{
u
:
x
(
u
)
∈
D
}
is the new region over which
u
is integrated. If we think of a function
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 Spring '07
 Enright
 Math, Integrals, Sin, Spherical coordinate system, Multiple integral, Polar coordinate system

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