cylindrical coordinate system and its transformation to cartesian or rectangular coordinate system

A cylindrical coordinate system is used for cylindrical symmetrical problems (examples, cables, machine rotor etc.). This system does not have axes like cartesian coordinate system.

Any point P ( r, Θ ,z ) is represented by the intersection of three  mutually orthogonal surfaces.

where r the radius of a cylinder.

Θ is the azimuthal angle and

z is the constant plane is same as in cartesian coordinate system.

Representation of unit vectors in cylindrical coordinate system:

The three unit vectors can be represented as ri, Θj, zk.

ri is directed radially outward, normal to the cylinder surface at the  point of consideration that is along increasing r direction.

Θj is perpendicular to Θ = constant plane and is directed outward in the advance Θ direction.

z is perpendicular to z= constant  plane like cartesian coordinate system that is increasing z direction.

Unit vectors ri, Θj, zk  are mutually perpendicular to each other.

Properties of cylindrical unit vectors:

  • cross product of cylindrical unit vectors:

ri x Θj = zk

Θj x zk = ri

zk x  ri = Θj

  • dot product of cylindrical unit vectors:

ri.ri= Θj. Θj = zk.zk =1

ri.Θj= Θj.zk= zk. Θj=0

ri and Θj vary with coordinate Θ and  their direction changes as Θ changes.

This must be taken into consideration while   doing integration or differentiation process.

Transformation of cartesian coordinates or rectangular coordinates to cylindrical coordinates:

The cylindrical coordinates can be transformed to cartesian or rectangular coordinates  and vice versa and the relations will be:

x = rcos Θ

y = rsin Θ

r = square root of (x2 + y2)

Θ = tangent inverse(y/x)

z = z

Two step process is required  for transformation of a vector function from one coordinate system to an other.

First Step: Find vector components.

Second Step: Then find dot  product  between unit vectors.

Reference: This article is referred from my authored book “concepts of electromagnetic field theory” having ISBN 978-81-272-5245-8. Try to make the figures for cylindrical coordinate system. In case of any doubt in this article or any other EMFT or physics related article, kindly post in the comment section.

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