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Key words: Change of variables, Jacobian for implicit transformation,
hyperbolic coordinates, rotation
Key concepts: Know how to deal with implicit transformations
19.1 Classical coordinate systems
Cylindrical coordinates.
In cylindrical coordinates, a point (
x,y,z
) in
R
3
is repre
sented by a triple (
ρ,φ,z
) where
ρ
denotes the distance to the projection of the point
on the
xy
plane to the origin, namely
ρ
=
p
x
2
+
y
2
,
φ
denotes the angle between the
line from the origin to the point (
x,y,
0) and the
x
axis, and
z
is the height of the point
(
x,y,z
) above the
xy
plane. Therefore with some geometry we obtain
x
=
ρ
cos
φ
y
=
ρ
sin
φ
z
=
z
as the equations representing the transformation from Cartesian to cylindrical coordinates.
The Jacobian determinant for this transformation is easily seen to be
ρ
. We insist there
fore that
ρ >
0, and also 0
≤
φ <
2
π
.
Example 1.
Find the volume of the region bounded above by the paraboloid
z
=
x
2
+
y
2
,
on the sides by the cylinder
x
2
+
y
2
= 1 and below by the
xy
plane.
Solution.
The volume is
R R R
D
1
dV
where
D
is the region in question. The region
D
can be represented easily in cylindrical coordinates as
D
=
{
(
ρ,φ,z
) : 0
≤
φ <
2
π,
0
< ρ
≤
1
,
0
≤
z
≤
ρ
2
}
since
z
≤
x
2
+
y
2
represents the region below the paraboloid. Therefore
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.
 Spring '07
 Enright
 Math, Transformations

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