Math 200, Spring 2010
Handout
#21
Section 15-5
Change of Variables
Transformation Φ from the
uv
-Plane to the
xy
-Plane
A
transformation
or
mapping
is a function
:
X
Y
Φ
→
from a set
X
(the domain) to a set
Y
.
Suppose
2
:
D
Φ
→
ℝ
is the transformation
( ) ( ) ( )
( )
,
,
,
,
u v
u v
u v
φ
ψ
Φ
=
from the
uv
-plane to the
xy
-plane.
Here
and
are the component functions and
( ) ( )
,
,
,
x
u v
y
u v
=
=
. Transformation
Φ
is a function
whose domain and range are both subsets of
2
ℝ
.
If
( ) ( )
0
0
0
0
,
,
u v
x y
Φ
=
, then the point
( )
0
0
,
x y
is called the
image
of the point
( )
0
0
,
u v
.
Φ
maps a region
D
in the
uv
-plane into a region
R
in the
xy
-plane called the
image of
D
.
If no two points have the same image,
Φ
is called
one-to-one
.
If
Φ
is one-to-one, then it has an inverse
transformation from the
xy
-plane to the
uv
-plane defined by
( )
1
( , )
,
x y
u v
−
Φ
=
.
1.
Described the image of a rectangle
[ ] [ ]
1
2
1
2
,
,
r r
θ θ
×
under the polar coordinate
transformation
2
2
:
Φ
→
ℝ
ℝ
defined by
( ) ( )
,
cos ,
sin
r
r
r
θ
Φ
=
.
2.
Suppose
D
is a square bounded by the lines
0,
1,
0,
1
u
u
v
v
=
=
=
=
.
Find the image
R
of the set
D
under the
transformation
( )
2
:
,
1
x
v y
u
v
Φ
=
=
+
.