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# Lesson 6: Polar, Cylindrical, and Spherical coordinates

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Lesson 6: Polar, Cylindrical, and Spherical coordinates
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Section 9.7Polar, Cylindrical, and Spherical CoordinatesMath 21aFebruary 15, 2008AnnouncementsNo class Monday 2/18. No oﬃce hours Tuesday 2/19.Yes oﬃce hours Wednesday 2/20 2–4pm SC 323.OutlineWhy diﬀerent coordinate systems?Polar CoordinatesCylindrical CoordinatesSpherical CoordinatesWhy diﬀerent coordinate systems?The dimension of space comes from natureThe measurement of space comes from usDiﬀerent coordinate systems are diﬀerent ways to measurespaceOutlineWhy diﬀerent coordinate systems?Polar CoordinatesCylindrical CoordinatesSpherical CoordinatesPolar CoordinatesConversion from polar tocartesian (rectangular)x = r cos θy = r sin θrConversion from cartesian toyθpolar:xr =x 2 + y 2xyycos θ =sin θ =tan θ =rrxSolutionWe have√√ −1x =2 cos (5π/4) =2 √ = −12√√ −1y =2 sin (5π/4) =2 √ = −12Example: Polar to RectangularExampleFind the rectangular coordinates of the point with polar√coordinates ( 2, 5π/4).Example: Polar to RectangularExampleFind the rectangular coordinates of the point with polar√coordinates ( 2, 5π/4).SolutionWe have√√ −1x =2 cos (5π/4) =2 √ = −12√√ −1y =2 sin (5π/4) =2 √ = −12Solution√√We have r =3 + 1 =4 = 2, and√31cos θ =sin θ =22This is satisﬁes by θ = π/6.Example: Rectangular to PolarExampleFind the polar coordinates of the point with rectangular√coordinates ( 3, 1).Example: Rectangular to PolarExampleFind the polar coordinates of the point with rectangular√coordinates ( 3, 1).Solution√√We have r =3 + 1 =4 = 2, and√31cos θ =sin θ =22This is satisﬁes by θ = π/6.Worksheet #1–#5Document Outline
• Announcements
• Why different coordinate systems?
• Polar Coordinates
• Cylindrical Coordinates
• Spherical Coordinates

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