Calculus II - 17

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Stewart Calculus Section 10.3

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  • Calculus II - 17

    1. 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. 2. Polar coordinates system:O - pole (or origin)Polar axis: a (horizontal) ray starting at OAny point P is represented by ( , ) . P r O Polar axis
    3. 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 hasmany representations.
    4. 4. Connection between polar and Cartesiancoordinates system: = , = = + , =Converting polar coordinates to Cartesiancoordinates is easy, while representing thepoint with Cartesian coordinates in terms ofpolar coordinates is harder.
    5. 5. = , = = + , =Ex: Convert ( , / ) from polar to Cartesiancoordinates, convert ( , ) from Cartesianto polar coordinates.
    6. 6. = , = = + , =Ex: Convert ( , / ) from polar to Cartesiancoordinates, convert ( , ) from Cartesianto polar coordinates. ( , / ) ( , )
    7. 7. = , = = + , =Ex: Convert ( , / ) from polar to Cartesiancoordinates, convert ( , ) from Cartesianto polar coordinates. ( , / ) ( , ) ( , / ) ( , )
    8. 8. = , = = + , =Ex: Convert ( , / ) from polar to Cartesiancoordinates, convert ( , ) from Cartesianto polar coordinates. ( , / ) ( , ) ( , / ) ( , ) not ( , / )!
    9. 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. 10. Ex: =
    11. 11. Ex: =
    12. 12. Ex: =
    13. 13. Ex: = + (cardioid)
    14. 14. Ex: = + (cardioid)
    15. 15. Ex: = + (cardioid)
    16. 16. Ex: =
    17. 17. Ex: =
    18. 18. Ex: =
    19. 19. Ex: =
    20. 20. Ex: = ( . )+ ( . )
    21. 21. Ex: =
    22. 22. Ex: = ( . )+ ( )

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