Cylindrical Coordinates

Cylindrical Coordinates

Transforms

The forward and reverse coordinate transformations are

z

! r

^ z ^ " ^ !

z

!= x +y

2

2

" = arctan ( y, x ) z= z

x = ! cos" y = ! sin " z= z

x " y

where we formally take advantage of the two argument arctan function to eliminate quadrant confusion.

Unit Vectors

The unit vectors in the cylindrical coordinate system are functions of position. It is convenient to express them in terms of the cylindrical coordinates and the unit vectors of the rectangular coordinate system which are not themselves functions of position. ! ˆ ˆ ! x x + yy ˆ ˆ ˆ != = = x cos" + y sin " ! !

ˆ z ˆ ˆ " = ˆ # ! = $ x sin " + ˆ cos " y z ˆ ˆ=z

Variations of unit vectors with the coordinates

Using the expressions obtained above it is easy to derive the following handy relationships:

ˆ !" =0 !" ˆ !" ˆ ˆ ˆ = $ x sin # + y cos # = # !# ˆ !" =0 !z

ˆ !" =0 !# ˆ !" ˆ ˆ ˆ = $ x cos " $ y sin " = $ # !" ˆ !" =0 !z

!ˆ z =0 !" !ˆ z =0 !# ˆ !z =0 !z

Path increment

! We will have many uses for the path increment dr expressed in cylindrical coordinates: ! ˆ z ˆ ˆ z dr = d (! ! + zˆ ) = ! d! + ! d! + ˆ dz + zdˆ z

$ "! ˆ ˆ ˆ $ "z ˆ ˆ ˆ "! "! ' "z "z ' ˆ ˆ = ! d! + !& d! + d# + dz ) + zdz + z & d! + d# + dz) "! "# "z ( "! "# "z ( % % ˆ! d# + zdz ˆ = ! d! + # ˆ

Time derivatives of the unit vectors

We will also have many uses for the time derivatives of the unit vectors expressed in cylindrical coordinates:

ˆ ˆ ˙ ˆ ˙ "! ! + "! # + "! z = ## ˆ ˙ != ˙ ˆ˙ "! "# "z ˆ ˆ ˆ ˙ "# ˙ "# ˙ "# ˆ ˆ˙ ˙ #= !+ #+ z = $!# "! "# "z ˆ ˆ z ˙ ˙ = "z ! + "ˆ # + "z z = 0 ˙ z ˆ ˙ "! "# "z

Velocity and Acceleration

The velocity and acceleration of a particle may be expressed in cylindrical coordinates by taking into account the associated rates of change in the unit vectors: ! ! ˙ ˙ ˆ ˆ˙ z ˆ ˆ˙ ˆ ˙ ˆ ˙ ˆ˙ v = r = ! ! + ! ! + ˙ z + zz = !! + " !" + zz ! ˆ˙ ˆ ˙ ˆ˙ v = ! ! + "! " + zz

! ! ˙ ˙ ˙ ˆ ˙ ˆ ˙˙ ˆ ˙ ˆ ˙ ˙ ˆ ˙ ˙ ˙ ˆz ˆ a = v = !!...