S E C T I O N
13.7
Cylindrical and Spherical Coordinates
(ET Section 12.7)
411
SOLUTION
A point
P
on the parabola
C
has the form
P
=
x
0
,
ax
2
0
,
c
, hence the parametric equations of the line
through the origin and
P
are
x
=
tx
0
,
y
=
tax
2
0
,
z
=
tc
To find a direct relation between
xy
and
z
we notice that
yz
=
tax
2
0
ct
=
ac
(
tx
0
)
2
=
acx
2
Now, defining new variables
z
=
u

v
and
y
=
u
+
v
. This equation becomes
(
u
+
v)(
u

v)
=
acx
2
u
2

v
2
=
acx
2
⇒
u
2
=
acx
2
+
v
2
This is the equation of an elliptic cone in the variables
x
,
v
,
u
. We, thus, showed that the cone on the parabola
C
is
transformed to an elliptic cone by the transformation (change of variables)
y
=
u
+
v
,
z
=
u

v
,
x
=
x
.
13.7 Cylindrical and Spherical Coordinates
(ET Section 12.7)
Preliminary Questions
1.
Describe the surfaces
r
=
R
in cylindrical coordinates and
ρ
=
R
in spherical coordinates.
SOLUTION
The surface
r
=
R
consists of all points located at a distance
R
from the
z
axis. This surface is the cylinder
of radius
R
whose axis is the
z
axis. The surface
ρ
=
R
consists of all points located at a distance
R
from the origin.
This is the sphere of radius
R
centered at the origin.
2.
Which statement about the cylindrical coordinates is correct?
(a)
If
θ
=
0, then
P
lies on the
z
axis.
(b)
If
θ
=
0, then
P
lies in the
xz
plane.
SOLUTION
The equation
θ
=
0 defines the halfplane of all points that project onto the ray
θ
=
0, that is, onto the
nonnegative
x
axis. This half plane is part of the
(
x
,
z
)
plane, therefore if
θ
=
0, then
P
lies in the
(
x
,
z
)
plane.
z
y
x
The halfplane
q
=
0
For instance, the point
P
=
(
1
,
0
,
1
)
satisfies
θ
=
0, but it does not lie on the
z
axis. We conclude that statement (b) is
correct and statement (a) is false.
3.
Which statement about spherical coordinates is correct?
(a)
If
φ
=
0, then
P
lies on the
z
axis.
(b)
If
φ
=
0, then
P
lies in the
xy
plane.
SOLUTION
The equation
φ
=
0 describes the nonnegative
z
axis. Therefore, if
φ
=
0,
P
lies on the
z
axis as stated in
(a). Statement (b) is false, since the point
(
0
,
0
,
1
)
satisfies
φ
=
0, but it does not lie in the
(
x
,
y
)
plane.
4.
The level surface
φ
=
φ
0
in spherical coordinates, usually a cone, reduces to a halfline for two values of
φ
0
. Which
two values?
SOLUTION
For
φ
0
=
0, the level surface
φ
=
0 is the upper part of the
z
axis. For
φ
0
=
π
, the level surface
φ
=
π
is
the lower part of the
z
axis. These are the two values of
φ
0
where the level surface
φ
=
φ
0
reduces to a halfline.
5.
For which value of
φ
0
is
φ
=
φ
0
a plane? Which plane?
SOLUTION
For
φ
0
=
π
2
, the level surface
φ
=
π
2
is the
xy
plane.
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412
C H A P T E R
13
VECTOR GEOMETRY
(ET CHAPTER 12)
z
y
P
P
x
π
2
π
2
Exercises
In Exercises 1–4, convert from cylindrical to rectangular coordinates.
1.
(
4
,
π
,
4
)
SOLUTION
By the given data
r
=
4,
θ
=
π
and
z
=
4. Hence,
x
=
r
cos
θ
=
4 cos
π
=
4
·
(

1
)
=

4
y
=
r
sin
θ
=
4 sin
π
=
4
·
0
z
=
4
⇒
(
x
,
y
,
z
)
=
(

4
,
0
,
4
)
2.
2
,
π
3
,

8
SOLUTION
We are given that
(
r
,
θ
,
z
)
=
(
2
,
π
3
,

8
)
. Hence,
x
=
r
cos
θ
=
2 cos
π
3
=
2
·
1
2
=
1
y
=
r
sin
θ
=
2 sin
π
3
=
2
√
3
2
=
√
3
z
=

8
⇒
(
x
,
y
,
z
)
=
1
,
√
3
,

8
3.
0
,
π
5
,
1
2
SOLUTION
We have
r
=
0,
θ
=
π
5
,
z
=
1
2
. Thus,
x
=
r
cos
θ
=
0
·
cos
π
5
=
0
y
=
r
sin
θ
=
0
·
sin
π
5
=
0
z
=
1
2
⇒
(
x
,
y
,
z
)
=
0
,
0
,
1
2
4.
1
,
π
2
,

2
SOLUTION
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 Spring '08
 Hubscher
 Equations, Cartesian Coordinate System, Parametric Equations, Cos, Polar coordinate system

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