Bob Brown, CCBC Dundalk
Math 253 Calculus 3, Chapter 11 Section 7
In Calculus 2, you saw that some two-dimensional graphs are easier to represent in polar
coordinates than in rectangular coordinates. A similar situation exists for surfaces in
space. In this section, we will briefly study two alternative space-coordinate systems.
The Cylindrical Coordinate System
Cylindrical coordinates are especially convenient for representing cylindrical surfaces
and surfaces of revolution where the z-axis is the axis of symmetry.
In a cylindrical coordinate system, a point P in space is represented by an ordered triple
is a polar representation of the projection of P in the
is the
from r ,  to P
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Bob Brown, CCBC Dundalk
Math 253 Calculus 3, Chapter 11 Section 7
 

Exercise 1a: Plot the following point, P, given in cylindrical coordinates.  6 ,  , 2 
4 

z
y
x
 

Exercise 1b (Section 11.7 #4): Convert  6 ,  , 2  from cylindrical coordinates to
4 

rectangular coordinates.


Exercise 2 (Section 11.7 #12): Convert 2 3 ,  2 , 6 from rectangular coordinates to
cylindrical coordinates.
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Bob Brown, CCBC Dundalk
Math 253 Calculus 3, Chapter 11 Section 7
3
The Spherical Coordinate System
In the spherical coordinate system, each point is represented by an ordered triple: the first
coordinates is a distance, and the second and third coordinates are angles.
For example, assume that
the earth is spherical with
radius 3959 miles and that
the earth’s center is
designated as the origin of
the spherical coordinate
system. Assume also that
the positive branch of the
x-axis extends from the
center of the earth through
the point where the Prime
Meridian intersects the
Equator.
Dundalk’s longitudinal and
latitudinal coordinates are
approximately
39.3°N, 76.5°W. (See the
red dot to the right.)
Dundalk’s spherical
coordinates would be
3959 ,  76.5 , 50.7  .


Bob Brown, CCBC Dundalk
Math 253 Calculus 3, Chapter 11 Section 7
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In a spherical coordinate system, a point P is represented by an ordered triple  , ,   ,
where
is the distance between
is the same angle used in cylindrical coordinates for r  0
is the angle between the
Note that the
and
coordinates are
and the
.
Bob Brown, CCBC Dundalk
Math 253 Calculus 3, Chapter 11 Section 7
Converting Rectangular Coordinates to Spherical Coordinates
Exercise 3 (Section 11.7 #34): Convert  1, 2 ,1 from rectangular coordinates to
spherical coordinates.
Converting Spherical Coordinates to Rectangular Coordinates


Exercise 4 (Section 11.7 #40): Convert  6 ,  ,  from spherical coordinates to
2

rectangular coordinates.
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