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11.7 Day 1 Cylindrical Coordinates

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Comparing Cartesian and cylindrical coordinates

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

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Conversion formulas from cylindrical to rectangular coordinates

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

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

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Solution to example 1

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

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

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

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

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Solution to 3a From the preceding section, you know that x 2 + y 2 = 4z 2 is a double napped cone with its axis along the z-axis as shown. If you replace x 2 + y 2 with r 2, the equation in cylindrical Coordinates is r 2 = 4z 2 x 2 + y 2 = 4z 2 Rectangular equation r 2 = 4z 2 Cylindrical equation

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

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Solution to 3b

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Example 4 Find an equation in rectangular coordinates for the surface represented by the cylindrical equation: Identify the surface r 2 cos2 θ +z 2 +1 = 0

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y z x

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Changing between coordinates on the TI 89 Press 2 nd 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

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

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

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