15.9: CHANGE OF VARIABLES IN MULTIPLE
INTEGRALS
KIAM HEONG KWA
1.
C
1
Transformations and Their Jacobians
By a transformation
or change of variables
T
:
S
→
R
is meant a
function whose domain
S
and range
R
are both subsets of
R
2
. For ease
of distinction, we denote the space in which
R
lies as the
uv
plane,
while the space in which
S
lies as the
xy
plane. It is then customary
to write
T
(
u, v
) = (
x
(
u, v
)
, y
(
u, v
))
,
(
u, v
)
∈
S,
where
x
(
u, v
) and
y
(
u, v
) denote the first and the second components
of
T
.
The point (
x
(
u, v
)
, y
(
u, v
)) is referred to as the image of the
point (
u, v
), while the range
R
=
T
(
S
) is called the image of
S
un
der the transformation
T
.
Such a transformation
T
is said to be
continuously differentiable
or
C
1
provided
x
u
,
x
v
,
y
u
, and
y
v
are well
defined and continuous.
We shall always assume that all transformations are
C
1
.
For our purpose in this section, we also require that a transformation
T
be injective
or onetoone
in the sense that no two points in
S
has the
same image. That is to say, if
T
(
u
1
, v
1
) =
T
(
u
2
, v
2
), then it is necessary
that (
u
1
, v
1
) = (
u
2
, v
2
). In this case, there is an inverse transformation
T

1
:
R
→
S
from the
xy
plane into the
uv
plane such that
T

1
◦
T
(
u, v
) = (
u, v
) and
T
◦
T

1
(
x, y
) = (
x, y
)
for all (
u, v
)
∈
S
and all (
x, y
)
∈
R
.
Next, we define the Jacobian
of
T
as
(1.1)
∂
(
x, y
)
∂
(
u, v
)
=
x
u
x
v
y
u
y
v
=
x
u
y
v

x
v
y
u
.
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 Fall '10
 Kwa
 Calculus, Geometry, Integrals, Transformations, Sets, xy plane, KIAM HEONG KWA, u∗, yu yv

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