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Calculus
Stewart
Dr. Berg
Summer ‘09
Page 1
11.3
11.3
Polar Coordinates
The position of a point in a plane is determined by two numbers. We normally use
Cartesian coordinate systems, but there are others. Many curves are more easily described
using polar coordinates that give the angle of a ray through the point and the distance
from the origin.
Definition
Fix a point called the
pole
and extend a ray to the right, which we call the
pole
axis
. The
polar coordinates
of a point are a pair of real numbers (
r
,
θ
) where
is the
angle in radians between the polar axis and a ray from the pole through the point and
r
is
the distance.
It is convenient to define (0,
) to be the pole for any angle
. It is also convenient
to allow
r
to be negative and take that to mean going in the opposite direction.
(
r
,
θ
)
(–
r
,
θ
)
Notice that, unlike Cartesian coordinates, polar coordinates are not unique. The
origin is represented by (0,
) where
can be any angle. Also, a point with coordinates
(
r
,
) can also be represented by
(
r
,
+
2
n
π
)
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This note was uploaded on 06/06/2010 for the course M 408 taught by Professor Hodges during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Hodges

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