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Cylindrical Coordinates

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Cylindrical coordinates are an extension of two-dimensional polar coordinates to three-dimensions.
cylindrical coordinate system

With cylindrical coordinates, the usual x- and y-coordinates of a point in the Cartesian plane are replaced by polar coordinates. A point with P = (x, y, z) has cylindrical coordinates P = (r, θ, z) where (r, θ) are polar coordinates of (x, y).

Use of Cylindrical Coordinates

Primarily used in physics, the cylindrical coordinate system (formally called the circular cylindrical or circular polar coordinate system) eases calculations involving cylinders or cylindrical symmetries (Hassani, 2009). For example:

  • Pipe flows with no-slip conditions at the wall drive pipe flow and require the use of cylindrical coordinates (Orlandi, 2012).
  • A magnetic field, generated by a current flowing in a long straight wire, is more conveniently expressed in cylindrical form (Rogawski, 2007).

Converting from Rectangular (Cartesian) to Cylindrical Coordinates

To convert, replace the x and y coordinates with the polar coordinates r and θ, using the following relations for x and y:

  • r = √(x2 + y2)
  • tanθ = y/x

The z-coordinate remains unchanged.

Example question: Convert (√3, 1, 4) from rectangular to cylindrical coordinates.

Solution: Take each point (x, y, z) one at a time and convert (√3, 1, 4) using the above relations, to get:

  • r = √(x2 + y2) = √(3 + 1)
  • θ = tan-1(y/x) = tan-1(1/√3) = π/6
  • z = 4.

Putting those together, we get (2, π/6, 4).


Converting from Cylindrical to Rectangular Coordinates

Use the relations:

  • x = r cosθ
  • y = r sinθ.

The z-coordinate remains unchanged.

Example question: Convert the cylindrical coordinates (3, π/3, -4) to rectangular coordinates.

Solution:

  • x = r cosθ = 3 cos (π/3) = 3(½) = 3/2
  • y = r sinθ = 3 sin (π/3) = 3 (√3/2) = (3√3)/2

The z-coordinate remains unchanged, giving:
(3/2, (3√3)/2, -4).


References

Glahodny, G. Section 13.9: Cylindrical and Spherical Coordinates. Retrieved December 2, 2020 from: https://www2.math.tamu.edu/~glahodny/
Hassani, S. (2009). Mathematical Methods For Students of Physics and Related Fields. Springer.
Orlandi, P. (2012). Fluid Flow Phenomena: A Numerical Toolkit (Fluid Mechanics and Its Applications Book 55), Kindle Edition. Springer.
Rogawski, J. (2007). Multivariable Calculus: Early Transcendentals. W. H. Freeman.

CITE THIS AS:
Stephanie Glen. "Cylindrical Coordinates" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/cylindrical-coordinates/
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