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# Lesson 6: Polar, Cylindrical, and Spherical coordinates

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"The fact that space is three-dimensional is due to nature. The way we measure it is due to us." Cartesian coordinates are one familiar way to do that, but other coordinate systems exist which are more useful in other situations.

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### Lesson 6: Polar, Cylindrical, and Spherical coordinates

1. 1. Section 9.7 Polar, Cylindrical, and Spherical Coordinates Math 21a February 15, 2008 Announcements No class Monday 2/18. No oﬃce hours Tuesday 2/19. Yes oﬃce hours Wednesday 2/20 2–4pm SC 323.
2. 2. Outline Why diﬀerent coordinate systems? Polar Coordinates Cylindrical Coordinates Spherical Coordinates
3. 3. Why diﬀerent coordinate systems? The dimension of space comes from nature The measurement of space comes from us Diﬀerent coordinate systems are diﬀerent ways to measure space
4. 4. Outline Why diﬀerent coordinate systems? Polar Coordinates Cylindrical Coordinates Spherical Coordinates
5. 5. Polar Coordinates Conversion from polar to cartesian (rectangular) x = r cos θ y = r sin θ r Conversion from cartesian to y θ polar: x r= x2 + y2 x y y cos θ = sin θ = tan θ = r r x
6. 6. Example: Polar to Rectangular Example Find the rectangular coordinates of the point with polar √ coordinates ( 2, 5π/4).
7. 7. Example: Polar to Rectangular Example Find the rectangular coordinates of the point with polar √ coordinates ( 2, 5π/4). Solution We have √ √ −1 x= 2 √ = −1 2 cos (5π/4) = 2 √ √ −1 y = 2 sin (5π/4) = 2 √ = −1 2
8. 8. Example: Rectangular to Polar Example Find the polar coordinates of the point with rectangular √ coordinates ( 3, 1).
9. 9. Example: Rectangular to Polar Example Find the polar coordinates of the point with rectangular √ coordinates ( 3, 1). Solution √ √ We have r = 3+1= 4 = 2, and √ 3 1 cos θ = sin θ = 2 2 This is satisﬁes by θ = π/6.
10. 10. Worksheet #1–#5
11. 11. Outline Why diﬀerent coordinate systems? Polar Coordinates Cylindrical Coordinates Spherical Coordinates
12. 12. Cylindrical Coordinates Just add the vertical dimension Conversion from cylindrical to cartesian (rectangular): x = r cos θ y = r sin θ z =z Conversion from cartesian to cylindrical: r = x2 + y2 x y y cos θ = sin θ = tan θ = r r x z =z
13. 13. Worksheet #6–#8
14. 14. Outline Why diﬀerent coordinate systems? Polar Coordinates Cylindrical Coordinates Spherical Coordinates
15. 15. Spherical Coordinates like the earth, but not exactly Conversion from spherical to cartesian (rectangular): x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ Conversion from cartesian to spherical: r= x2 + y2 ρ = x2 + y2 + z2 x y y cos θ = sin θ = tan θ = Note: In this picture, r should r r x be ρ. z cos ϕ = ρ
16. 16. Examples Example Find the spherical coordinates of the point with rectangular √ √ coordinates ( 2, −2, 3).
17. 17. Examples Example Find the spherical coordinates of the point with rectangular √ √ coordinates ( 2, −2, 3). Answer 1 1 3, 2π − arccos √ , arccos √ 3 3
18. 18. Examples Example Find the spherical coordinates of the point with rectangular √ √ coordinates ( 2, −2, 3). Answer 1 1 3, 2π − arccos √ , arccos √ 3 3 Example Find the rectangular coordinates of the point with spherical coordinates (2, π/6, 2π/3).
19. 19. Examples Example Find the spherical coordinates of the point with rectangular √ √ coordinates ( 2, −2, 3). Answer 1 1 3, 2π − arccos √ , arccos √ 3 3 Example Find the rectangular coordinates of the point with spherical coordinates (2, π/6, 2π/3). Answer √ √ √ √ 3 3 3 1 −1 3 3 2· · ,2 · · ,2 · = , , −1 2 2 2 2 2 2 2
20. 20. Worksheet #9–#10