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

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

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

Solution to example 1

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

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

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

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

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

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

Solution to 3b

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

z y x

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

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

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