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Cylindrical Coordinate


A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

The
origin of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis.

The axis is variously called the
cylindrical or longitudinal axis, to differentiate it from the polar axis, which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction.
The distance from the axis may be called the
radial distance or radius, while the angular coordinate is sometimes referred to as the angular position or as the azimuth. The radius and the azimuth are together called the polar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position or axial position.

Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metal cylinder, and so on.

The relations used for the conversion of the coordinates of the point from the Cartesian coordinate system to the cylindrical coordinate system are:


In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.
The line element is
.
The volume element is
.
The gradient is
 

A vector in the cylindrical coordinate can also be written as:

A = ayAy + aøAø + azAzØ is the angle started from x axis.

The differential length in the cylindrical coordinate is given by:

dl = ardr + aø ∙ r ∙  + azdz

The differential area of each side in the cylindrical coordinate is given by:
dsy = r ∙ dø ∙ dz
dsø = dr ∙ dz
dsz = r ∙ dr ∙ dø

The differential volume in the cylindrical coordinate is given by:
dv = r ∙ dr ∙ dø ∙ dz

Example 1: Convert the point (6, 8, 4.5) in Cartesian coordinate system to cylindrical coordinate system.
Solution:

So the equivalent cylindrical coordinates are (10, 53.1, 4.5)

Example 2: Convert (1/2, √(3)/2, 5) to cylindrical coordinates.
Solution:
r2 = x2 + y2 = (1/2)2 + (√(3)/2)2 = 1
and
tanθ = y/x = √(3)
Thus θ = π/3 and r = 1. Thus cylindrical coordinates are (1, π/3, 5).

Example 3: What is the equation in cylindrical coordinates of a cone x2 + y2 = z2?
Solution:
Replacing x2 + y2 by r2, we obtain r2 = z2 which usually gives us r= ± z. Since z can be any real number it is enough to write r = z. Thus in cylindrical coordinates this cone is r = z.

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