Cylindrical and Spherical
Coordinates
Representation and
Conversions
Representing 3D points in
Cylindrical Coordinates.
Recall polar representations in the plane
r

Representing 3D points in
Cylindrical Coordinates.
Cylindrical coordinates just adds a z-coordinate to the
polar coordinates (r,).
r

Representing 3D points in
Cylindrical Coordinates.
(r,,z)
r

r

Converting between rectangular
and Cylindrical Coordinates
No real surprises here!
(r,,z)
r

Cylindrical to rectangular
x  r cos( )
y  r sin( )
zz
Rectangular to Cylindrical
r

r 2  x2  y 2
y
tan( ) 
x
zz
Representing 3D points in Spherical
Coordinates
Spherical Coordinates are the 3D
analog of polar representations in
the plane.
We divide 3-dimensional space into
a set of concentric spheres centered at
the origin.
2. rays emanating outward from the origin
1.
Representing 3D points in Spherical
Coordinates
(x,y,z)

We start with a point (x,y,z) given
in rectangular coordinates.
Then, measuring its distance 
from the origin, we locate it on a
sphere of radius  centered at the
origin.
Next, we have to find a way to
describe its location on the sphere.
Representing 3D points in Spherical
Coordinates
We use a method similar to the
method used to measure latitude
and longitude on the surface of the
Earth.
We find the great circle that goes
through the “north pole,” the “south
pole,” and the point.
Representing 3D points in Spherical
Coordinates

We measure the latitude or polar
angle starting at the “north pole” in
the plane given by the great circle.
This angle is called . The range
of this angle is 0     .
Note:
all angles are measured in
radians, as always.
Representing 3D points in Spherical
Coordinates
We use a method similar to the
method used to measure latitude
and longitude on the surface of the
Earth.
Next, we draw a horizontal circle
on the sphere that passes through
the point.
Representing 3D points in Spherical
Coordinates
And “drop it down” onto the xyplane.
Representing 3D points in Spherical
Coordinates
We measure the latitude or azimuthal
angle on the latitude circle, starting at
the positive x-axis and rotating toward
the positive y-axis.
The range of the angle is
0    2 .
Angle is called .
Note that this is the same angle as the  in cylindrical coordinates!
Finally, a Point in Spherical
Coordinates!
( , ,)

Our designated point on the
sphere is indicated by the three
spherical coordinates ( ,  , ) --(radial distance, azimuthal angle,
polar angle).
Please note that this notation is not
at all standard and varies from
author to author and discipline to
discipline. (In particular, physicists
often use  to refer to the
azimuthal angle and  refer to the
polar angle.)
Converting Between Rectangular
and Spherical Coordinates
(x,y,z)
First note that if r is the usual cylindrical
coordinate for (x,y,z)
r

 z
we have a right triangle with
•acute angle ,
•hypotenuse , and What happens if
 is not acute?
•legs r and z.
It follows that
sin( ) 
r

cos( ) 
z

r
tan( ) 
z
Converting Between Rectangular
and Spherical Coordinates
(x,y,z)
r

Spherical to rectangular
 z
x  r cos( )   sin( ) cos( )
y  r sin( )   sin( ) sin( )
z   cos( )
Converting from Spherical to
Rectangular Coordinates
Rectangular to Spherical
(x,y,z)
r

 z
  x2  y 2  z 2
tan( ) 
y
x
x2  y 2
r
tan( )  
z
z
z
z
cos( )  

x2  y 2  z 2
Cylindrical and Spherical
Coordinates
Integration
Integration Elements:
Rectangular Coordinates
We know that in a Riemann Sum approximation for a
triple integral, a summand
f ( x , y , z )xi yi zi .
*
i
*
i
*
i
This computes the function value at some point in the little
“sub-cube” and multiplies it by the volume of the little
cube of length xi , width yi , and height zi .
f ( x , y , z ) xi yi zi .
*
i
*
i
*
i
function value
volume of the small
at a sampling point
cube
Integration Elements:
Cylindrical Coordinates
What happens when we consider small changes
r ,  , and z
in the cylindrical coordinates r, , and z?
We no longer get a
cube, and (similarly to
the 2D case with polar
coordinates) this affects
integration.
Integration Elements:
Cylindrical Coordinates
What happens when we consider small changes
r ,  , and z
in the cylindrical coordinates r, , and z?
Start with our previous
picture of cylindrical
coordinates:
r 
r 
Integration Elements:
Cylindrical Coordinates
What happens when we consider small changes
r ,  , and z
in the cylindrical coordinates r, , and z?
Start with our previous
picture of cylindrical
coordinates:
Expand the radius by a
small amount:
r
r
r
r+r
Integration Elements:
Cylindrical Coordinates
This leaves us with a thin cylindrical shell of inner radius r
and outer radius r+ r.
r
r+r
r
r+r
Integration Elements:
Cylindrical Coordinates
Now we consider the angle .


We want to increase it by a small amount .

Integration Elements:
Cylindrical Coordinates
This give us a “wedge.”
Combining this with the cylindrical shell created by the
change in r, we get

Integration Elements:
Cylindrical Coordinates
This give us a “wedge.”
Intersecting this wedge with the cylindrical shell created by
the change in r, we get
Integration Elements:
Cylindrical Coordinates
Finally , we look at a small vertical change  z .
Integration in Cylindrical Coordinates.
We need to find the volume of this little solid.
As in polar coordinates, we have the area of a horizontal
cross section is. . .
dA  r dr d 
Integration in Cylindrical Coordinates.
We need to find the volume of this little solid.
Since the volume is just the base times the height. . .
dV
 r dr d  dz
So . . .
 f (r, , z) r dr d  dz
S