Calculus Of One Real Variable – By Pheng Kim
Ving


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1. The Polar Coordinate System 

Fig. 1.1 Defining A Unique Point. 
To setup a polar coordinate system, select a point O
in the plane and draw a ray from O. See Fig. 1.2.
Conventionally for
convenience the ray is horizontal and to the right of O.
Definition 1.1 –
The Polar Coordinate System
A point O and a ray emanating from
it to the right form the polar coordinate system. The point O
is called the pole and 
Fig. 1.3 displays some points with their polar coordinates.

Fig. 1.2 Polar Coordinate System. 

Fig. 1.3 Some Points With Their Polar Coordinates. 
Angle
Negative
Distance

Fig. 1.4 Negative Distance. 
Infinitely Many
Polar Coordinates For A Point
Property 1.1 – Polar
Coordinates
For any real r > 0 and for all integer k: 
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2. Cartesian And Polar Coordinate Systems 
Property 2.1 – Cartesian
And Polar Coordinates


Fig. 2.1 Cartesian And Polar Coordinate Systems. 
Eqs. [2.1] express the Cartesian coordinates in terms of the
polar ones, and Eqs. [2.2] and [2.3] express the polar coordinates
in terms of the Cartesian ones. These equations can be used to transform the
coordinates of points from one coordinate
system to the other.
Example 2.1
Solution
EOS
Example 2.2
Solution
EOS
The quadrant where the point lies is important. The
coordinates of the point, Cartesian or polar, must mean that the point lies
in the correct quadrant.
Example 2.2 is the reverse of Example 2.1. The reverse task
maps a single ordered pair of Cartesian coordinates back to the
original ordered pair of polar coordinates plus maps it to infinitely many
other such ordered pairs. Of course the sign of the
distance depends on whether the given point lies on the ray determined by the
angle (positive) or its extension (negative).
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3. Polar Equations And Polar Curves 

Fig. 3.1 Polar Equation. 
Definition 3.1 –
Polar Equations And Polar Curves

A Cartesian equation generally represents some curve in the
Cartesian coordinate system. Similarly, a polar equation generally
represents some curve in the polar coordinate system.
Level Coordinate
Curves

Fig. 3.2 Level Coordinate Curves Of Polar System. 
Cartesian And
Polar Equations
Equations in Property 2.1 can be
utilized to transform the Cartesian representation of a curve into its polar
representation and
vice versa.
Example 3.1
Transform the Cartesian equation y = 2x – 1 of a line to a polar equation.
Solution
EOS
Example 3.2
Find the polar representation of the ellipse with Cartesian representation x^{2} + 3y^{2} = 5.
Solution
EOS
Example 3.3
Solution 1
EOS
Solution 2
EOS
Problems & Solutions 
Solution
Note
For part c, as r = 0, it's the origin, so there's no need to do any calculation.
Solution
Note
^{{p&s 1}} Problem & Solution 1.
3. Find the polar equation of each of the following
curves with the given Cartesian equation.
a. x = c.
b. x^{2}y + y^{3} = – 4.
Solution
4. Determine the Cartesian equation of each of the following curves with the given polar equation. Identify the curve.
Solution
Solution
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