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Section 13.7
Cylindrical and Spherical Coordinates
“NonRectangular Coordinate Systems in 3space”
In Calculus II, we considered the polar coordinate system to help inte
grate functions whose graphs were circular regions. In this section, we
consider two new coordinate systems for graphs in 3space.
1.
Cylindrical Coordinates
The Frst coordinate system we consider is a generalization of polar
coordinates  the basic idea is to take the polar coordinates in the
xy

plane and then simply add the
z
coordinate to determine the height
of a point. They are particularly useful when describing cylinders.
±ormally, we deFne the cylindrical coordinate system as follows.
Defnition 1.1.
The cylindrical coordinates of a point
P
in 3space
is deFned to be (
r, ϑ, z
) where (
r, ϑ
) are the polar coordinates of the
projection of
P
in the
xy
plane and
z
is the
z
coordinate of the plane
where
r
g
0 and 0
l
ϑ <
2
π
.
z
(r,theta)
Since cylindrical coordinates are so closely related to polar coordinates,
it is easy to convert from rectangular coordinates in 3space into cylin
drical and vice versa.
Result 1.2.
(
i
) The rectangular coordinates of the point (
r, ϑ, z
)
in 3 space are
x
=
r
cos (
ϑ
),
y
=
r
sin (
ϑ
) and
z
=
z
.
(
ii
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 Fall '08
 Stefanov
 Calculus

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