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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