MATH 220 SUPPLEMENTAL NOTES 7
CYLINDRICAL AND SPHERICAL COORDINATES
CYLINDRICAL COORDINATES
Cylindrical coordinates combine the polar coordinates in the xy-plane with the usual z-axis.
FACT: Cylindrical coordinates
represent a point P in space
by an ordered triple (r, , z)
in which (1) r and are
polar coordinates for the
vertical projection of P on
the xy-plane, and (2) z is
the rectangular vertical
coordinate. (See figure 1)
figure 1
EQUATIONS RELATING RECTANGULAR (x, y, z) COORDINATES AND
CYLINDRICAL (r, , z) COORDINATES
x = r cos y = r sin z=z
r2=x2+y2
SPECIAL EQUATIONS
r=a This is not just a circle in the xy-plane, it a right circular cylinder about the
z-axis.
=0 This equation describes the plane that contains the z-axis and makes an
angle 0 with the positive x-axis.
EXAMPLE 1: Convert (1, 3, 4) in rectangular coordinates to cylindrical
coordinates.
First of all, let us determine r.
SOLUTION:
Now to find .
z stays the same.
EXAMPLE 2: Convert (0, 0, 1) in rectangular coordinates to cylindrical
coordinates.
First determine r.
SOLUTION:
Now to determine .
Therefore, = /2.
The z value stays the same, so the point is (0, /2,1).
Convert
EXAMPLE 3:
in cylindrical coordinates to rectangular coordinates.
z stays the same, so I need to find the values for x and y. To do this,
SOLUTION: I will use r and .
EXAMPLE 4: Convert (2, /6, 6) in cylindrical coordinates to rectangular
coordinates.
z = 6, so all I need to find is the values for x and y.
SOLUTION:
EXAMPLE 5: Convert x 2 + y 2 = 5 into cylindrical coordinates.
Remember that r 2 = x 2 + y 2, so r 2 = 5 or
SOLUTION:
Convert
EXAMPLE 6:
into cylindrical coordinates.
SOLUTION:
EXAMPLE 7: Convert x 2 + y 2 + z 2 = 9 into cylindrical coordinates.
SOLUTION: x 2 + y 2 + z 2 = 9 r 2 + z 2 = 9
EXAMPLE 8: Convert r = -3sec to rectangular coordinates.
SOLUTION:
Convert
EXAMPLE 9:
to rectangular coordinates.
SOLUTION:
SPHERICAL COORDINATES
One thing that you should keep in
mind when working with
spherical coordinates is that is
never negative, is the angle that
OP makes with the positive z-
axis, therefore its range is [0, ],
and is measured as in cylindrical
coordinates. (See figure 2)
figure 2
FACT: Spherical coordinates represent a point P in space by the ordered triple ( ,
, ) in which
1. is the distance from P to the origin.
2. is the angle OP makes with the positive z-axis. ( 0 )
3. is the angle from cylindrical coordinates.
SPECIAL EQUATIONS
=a This equation describes a sphere of radius a centered at the origin.
This equation describes a single cone whose vertex lies at the origin and
=0 whose axis lies along the z-axis.
If = /2, then we have the xy-plane.
If > /2, then the cone = 0 opens downward.
EQUATIONS RELATING SPHERICAL COORDINATES TO CARTESIAN AND
CYLINDRICAL COORDINATES
z = cos r = sin x = r cos = sin cos y = r sin = sin sin
EXAMPLE 10: Convert (1, 3, 4) in rectangular coordinates to spherical
coordinates.
First, find .
SOLUTION:
Now find .
Now determine . To do this, I will use z = cos and solve for
.
EXAMPLE 11: Convert (0, 0, 1) in rectangular coordinates to spherical
coordinates.
SOLUTION:
Therefore, = /2.
1 = cos = 0.
So the point is (1, 0, /2).
Convert
EXAMPLE 12:
in cylindrical coordinates to spherical coordinates.
SOLUTION:
stays the same, so I need to find .
EXAMPLE 13: Convert (2, /6, 6) in cylindrical coordinates to spherical
coordinates.
SOLUTION:
= /6 = 30 o
Convert
EXAMPLE 14:
to rectangular coordinates, then cylindrical coordinates.
Rectangular coordinates
SOLUTION:
Cylindrical coordinates
Convert
EXAMPLE 15:
to rectangular coordinates, then cylindrical coordinates.
Rectangular coordinates
SOLUTION:
Cylindrical coordinates
EXAMPLE 16: Convert x 2 + y 2 = 5 into spherical coordinates.
SOLUTION:
EXAMPLE 17: Convert x 2 + y 2 + z 2 = 9 into spherical coordinates.
SOLUTION:
EXAMPLE 18: Convert r = -3sec into spherical coordinates.
SOLUTION: r = -3sec sin = -3sec = -3csc sec
Convert
EXAMPLE 19:
into spherical coordinates.
SOLUTION:
= 135 o
Here are the spherical equations.
=2
135 o 180o
Convert
EXAMPLE 20:
into cylindrical and rectangular coordinates.
SOLUTION:
The above equation is both the cylindrical form of the given
equation and the rectangular form of the given equation.
Convert
EXAMPLE 21:
into cylindrical and rectangular coordinates.
Let us convert to rectangular coordinates first.
SOLUTION:
x2+y2-z2=0
The rectangular equations are the following.
Now the cylindrical equations.
x 2 + y 2 - z 2 = 0 r 2 - z 2 = 0 r = -z or r = z, but r 0 and z 0
r = -z
You should realize from these examples that cylindrical coordinates work great for
equations of cylinders, and spherical coordinates work well for equations of spheres. You
will probably find that converting from rectangular coordinates to cylindrical coordinates
easier than converting from either rectangular or cylindrical coordinates to spherical. It is
just the nature of the beast (spherical coordinates) to be hard to convert to and from.
