Stewart Calculus Section 10.3

1. 10.3 Polar Coordinates
Cartesian coordinates: a point in the plane is
represented by the distances from two
perpendicular axes.
y
o x

2. Polar coordinates system:
O - pole (or origin)
Polar axis: a (horizontal) ray starting at O
Any point P is represented by ( , ) .
P
r
O Polar axis

3. Conventions:
Pole is represented by ( , ) for any .
is measured in the counter-clockwise
direction. ( , ) represents the same
point as ( , ).
(− , ) represents the same point as (, + ).
In polar coordinate system each point has
many representations.

4. Connection between polar and Cartesian
coordinates system:
= , =
= + , =
Converting polar coordinates to Cartesian
coordinates is easy, while representing the
point with Cartesian coordinates in terms of
polar coordinates is harder.

5. = , =
= + , =
Ex: Convert ( , / ) from polar to Cartesian
coordinates, convert ( , ) from Cartesian
to polar coordinates.

6. = , =
= + , =
Ex: Convert ( , / ) from polar to Cartesian
coordinates, convert ( , ) from Cartesian
to polar coordinates.
( , / ) ( , )

7. = , =
= + , =
Ex: Convert ( , / ) from polar to Cartesian
coordinates, convert ( , ) from Cartesian
to polar coordinates.
( , / ) ( , )
( , / ) ( , )

8. = , =
= + , =
Ex: Convert ( , / ) from polar to Cartesian
coordinates, convert ( , ) from Cartesian
to polar coordinates.
( , / ) ( , )
( , / ) ( , )
not ( , / )!

9. Polar curves.
Like in the Cartesian system, one polar
equation represents a curve in general.
Sometimes it is much simpler than the
corresponding representation in the
Cartesian system.
Ex: What is the curve described by = ?
What is the curve described by = ?

10. Ex: =

11. Ex: =

12. Ex: =

13. Ex: = + (cardioid)

14. Ex: = + (cardioid)

15. Ex: = + (cardioid)

16. Ex: =

17. Ex: =

18. Ex: =

19. Ex: =

20. Ex: = ( . )+ ( . )

21. Ex: =

22. Ex: = ( . )+ ( )