Cylindrical and Spherical Coordinates

1
Cylindrical and Spherical Coordinates

• Representation and Conversions

2
Representing 3D points in Cylindrical
Coordinates.
Recall polar representations in the plane
r
3
Representing 3D points in Cylindrical
Coordinates.
Cylindrical coordinates just adds a z-coordinate
to the polar coordinates (r,?).
r
4
Representing 3D points in Cylindrical
Coordinates.
Cylindrical coordinates just adds a z-coordinate
to the polar coordinates (r,?).
5
Representing 3D points in Cylindrical
Coordinates.
Cylindrical coordinates just adds a z-coordinate
to the polar coordinates (r,?).
6
Representing 3D points in Cylindrical
Coordinates.
Cylindrical coordinates just adds a z-coordinate
to the polar coordinates (r,?).
7
Representing 3D points in Cylindrical
Coordinates.
Cylindrical coordinates just adds a z-coordinate
to the polar coordinates (r,?).
8
Representing 3D points in Cylindrical
Coordinates.
Cylindrical coordinates just adds a z-coordinate
to the polar coordinates (r,?).
9
Representing 3D points in Cylindrical
Coordinates.
(r,?,z)
10
Converting between rectangular and Cylindrical
Coordinates
Cylindrical to rectangular
No real surprises here!
Rectangular to Cylindrical
11
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.
• rays emanating outward from the origin

12
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.
?
13
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.
14
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
?
15
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.
16
Representing 3D points in Spherical Coordinates
And drop it down onto the xy-plane.
17
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
Angle is called ?.
Note that this is the same angle as the ? in
cylindrical coordinates!
18
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.)
?
19
Converting Between Rectangular and Spherical
Coordinates

• First note that if r is the usual cylindrical
  coordinate for (x,y,z)
• we have a right triangle with
• acute angle ?,
• hypotenuse ?, and
• legs r and z.
• It follows that

(x,y,z)
r
?
z
?
What happens if ? is not acute?
20
Converting Between Rectangular and Spherical
Coordinates
(x,y,z)
r
Spherical to rectangular
?
z
?
21
Converting from Spherical to Rectangular
Coordinates
Rectangular to Spherical
(x,y,z)
r
?
z
?
22
Cylindrical and Spherical Coordinates

• Integration

23
Integration Elements Rectangular
Coordinates
We know that in a Riemann Sum approximation for a
triple integral, a summand 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 , width ,
and height .
24
Integration Elements Cylindrical
Coordinates
What happens when we consider small changes in
the cylindrical coordinates r, q, and z?
We no longer get a cube, and (similarly to the 2D
case with polar coordinates) this affects
integration.
25
Integration Elements Cylindrical
Coordinates
What happens when we consider small changes in
the cylindrical coordinates r, q, and z?
Start with our previous picture of cylindrical
coordinates
26
Integration Elements Cylindrical
Coordinates
What happens when we consider small changes in
the cylindrical coordinates r, q, and z?
Start with our previous picture of cylindrical
coordinates
r
Expand the radius by a small amount
r
rDr
27
Integration Elements Cylindrical
Coordinates
This leaves us with a thin cylindrical shell of
inner radius r and outer radius rD r.
r
rDr
r
rDr
28
Integration Elements Cylindrical
Coordinates
Now we consider the angle q. We want to
increase it by a small amount Dq.
29
Integration Elements Cylindrical
Coordinates
This give us a wedge. Combining this with
the cylindrical shell created by the change in r,
we get
30
Integration Elements Cylindrical
Coordinates
This give us a wedge. Intersecting this
wedge with the cylindrical shell created by the
change in r, we get
31
Integration Elements Cylindrical
Coordinates
Finally , we look at a small vertical change D z .
32
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. . .
33
Integration in Cylindrical Coordinates.
We need to find the volume of this little
solid. Since the volume is just the base times
the height. . .