Chapter 7: Cylindrical and Spherical Coordinates

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Chapter 7 Cylindrical and Spherical Coordinates

• Section 7.7

Written by Dr. Julia Arnold Associate Professor
of Mathematics Tidewater Community College,
Norfolk Campus, Norfolk, VA With Assistance from
a VCCS LearningWare Grant
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• In this lesson you will learn
• about cylindrical and spherical coordinates
• how to change from rectangular coordinates to
  cylindrical coordinates or spherical coordinates
• how to change from spherical coordinates to
  rectangular coordinates or cylindrical
  coordinates
• how to change from cylindrical coordinates to
  rectangular coordinates or spherical coordinates

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Polar Coordinates
The polar coordinates r (the radial coordinate)
and (the angular coordinate, often called the
polar angle) are defined in terms of Cartesian
Coordinates by
where r is the radial distance from the origin,
and is the counterclockwise angle from the
x-axis.
In terms of x and y,
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Cylindrical coordinates are a generalization of
two-dimensional polar coordinates to three
dimensions by superposing a height (z) axis.
As you can see, this coordinate system lends
itself well to cylindrical figures.
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Common Uses The most common use of cylindrical
coordinates is to give the equation of a surface
of revolution. If the z-axis is taken as the axis
of revolution, then the equation will not involve
theta at all. Examples A paraboloid of
revolution might have equation z r2. This is
the surface you would get by rotating the
parabola z x2 in the xz-plane about the z-axis.
The Cartesian coordinate equation of the
paraboloid of revolution would be z x2 y2. A
right circular cylinder of radius a whose axis is
the z-axis has equation r R. A a sphere with
center at the origin and radius R will have
equation r z2 R2. A right circular cone
with vertex at the origin and axis the z-axis has
equation z m r. As another kind of example, a
helix has the following equations r R,z a
theta. http//mathforum.org/dr.math/faq/formulas/
faq.cylindrical.html
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Solution
Work it out before you go to the next slide.
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Solution
You have two choices for r and infinitely many
choices for theta.
Thus the point can be represented
by non unique cylindrical coordinates. For
example
See picture on next slide.
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This graph was done using Win Plot in the two
different coordinate systems.
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The movie below the one above shows the points
represented by constant values of the second
coordinate as it varies from zero to 2 pi.
You can also view at this link http//www.tcc.edu
/faculty/webpages/JArnold/movies.htm
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Example 2  Identify the surface for each of the
following equations. (a) r 5 (b)
(c) z r  
Solution a. In polar coordinates we know that r
5 would be a circle of radius 5 units. By
adding the z dimension and allowing z to vary we
create a cylinder of radius 5.
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Example 2  Identify the surface for each of the
following equations. (a) r 5 (b)
(c) z r  
Solution b. This is equivalent to
which we know to be a sphere centered at
the origin with a radius of 10.
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Example 2  Identify the surface for each of the
following equations. (a) r 5 (b)
(c) z r  
Solution c. Since the radius equals the height
and the angle is any angle we get a cone.
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Spherical coordinates are a system of curvilinear
coordinates that are natural for describing
positions on a sphere or spheroid.
The ordered triple is
For a given point P in spherical coordinates
is the distance between P and the origin
is the same angle theta used in
cylindrical coordinates for
is the angle between the positive z-axis and the
line segment
(x,y,z)
P
z
O
The figure at right shows the Rectangular
coordinates (x,y,z) and The spherical coordinates
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Conversion Formulas Spherical to Rectangular
Rectangular to Spherical
Spherical to cylindrical ( )
Cylindrical to spherical ( )
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Example 3 A. Find a rectangular equation for the
graph represented by the cylindrical equation
B. Find an equation in spherical coordinates for
the surface represented by each of the
rectangular equations and identify the graph.
1.
2.
Answers follow
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Example 3 A. Find a rectangular equation for the
graph represented by the cylindrical equation
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Example 3
B. Find an equation in spherical coordinates for
the surface represented by each of the
rectangular equations and identify the graph.
1.
A double cone.
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Example 3
B. Find an equation in spherical coordinates for
the surface represented by each of the
rectangular equations and identify the graph.
2.
A sphere
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Review
Spherical to Rectangular
Rectangular to Spherical
Spherical to cylindrical ( )
Cylindrical to spherical ( )
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Do Exercises 7.7 in BB