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11.7 Day 1 Cylindrical Coordinates

11.7 Day 1 Cylindrical Coordinates. Comparing Cartesian and cylindrical coordinates. Note: these are just polar coordinates with a z coordinate (z is a vertical component). Conversion formulas from cylindrical to rectangular coordinates.

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11.7 Day 1 Cylindrical Coordinates

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  1. 11.7 Day 1 Cylindrical Coordinates

  2. Comparing Cartesian and cylindrical coordinates

  3. Note: these are just polar coordinates with a z coordinate (z is a vertical component)

  4. Conversion formulas from cylindrical to rectangular coordinates

  5. Converting between cylindrical coordinates and rectangular (Cartesian)Note: these formulas must be memorized

  6. Example 1 Convert the point (r, ө, z) = (4, 5π/6, 3) to rectangular coordinates.

  7. Solution to example 1

  8. Example 2 _ Convert the point (x, y, z) = (1, √3 , 2) to cylindrical coordinates.

  9. Cylindrical coordinates are usually more convenient for representing cylindrical surfaces as they often result in simpler equations.

  10. Vertical planes containing the z-axis and horizontal planes also have simple cylindrical coordinate equations

  11. Example 3 a Find an equation in cylindrical coordinates for the surfaces represented by the rectangular equation:

  12. Solution to 3a From the preceding section, you know that x2 + y2 = 4z2 is a “double napped” cone with its axis along the z-axis as shown. If you replace x2 + y2 with r2, the equation in cylindrical Coordinates is r2 = 4z2 x2 + y2 = 4z2 Rectangular equation r2 = 4z2 Cylindrical equation

  13. Example 3b Find an equation in cylindrical coordinates for the surfaces represented by the rectangular equation:

  14. Solution to 3b

  15. Example 4 Find an equation in rectangular coordinates for the surface represented by the cylindrical equation: Identify the surface r2cos2θ +z2 +1 = 0

  16. z y x

  17. Changing between coordinates on the TI 89 Press 2nd 5 (math) – 4 matrices – L Vecor ops To polar, to cynd To convert rectangular to Polar (2 D) or cylindrical (3D) [1,2] to Polar (to expressed with a triangle) [1,2,3] to Cylind

  18. Note: Homework do the assignment sheet plus the activity on the class website. This is a polar bear in rectangular form Bonus material on the next slides

  19. Need Help? If you are ever in need of assistance type in the following equation: See the next slide

  20. Help is on the way…

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