Cylindrical Coordinates
Cylindrical Coordinates
Cylindrical Coordinates
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<strong>Cylindrical</strong> <strong>Coordinates</strong><br />
Transforms<br />
The forward and reverse coordinate transformations are<br />
! = x 2 + y 2<br />
" = arctan( y, x)<br />
z = z<br />
x = ! cos"<br />
y = ! sin "<br />
z = z<br />
z<br />
z<br />
!<br />
r<br />
^<br />
z<br />
^<br />
!<br />
^<br />
"<br />
y<br />
where we formally take advantage of the two argument arctan<br />
function to eliminate quadrant confusion.<br />
x<br />
"<br />
Unit Vectors<br />
The unit vectors in the cylindrical coordinate system are functions of position. It is convenient to express them in terms of<br />
the cylindrical coordinates and the unit vectors of the rectangular coordinate system which are not themselves functions of<br />
position.<br />
ˆ ! =<br />
!<br />
! = x ˆ<br />
x + yˆ y<br />
!<br />
= ˆ x cos" + ˆ y sin "<br />
ˆ " = ˆ z # ˆ ! = $ x ˆ sin " + ˆ y cos "<br />
ˆ z = z ˆ<br />
Variations of unit vectors with the coordinates<br />
Using the expressions obtained above it is easy to derive the following handy relationships:<br />
"<br />
!# = 0<br />
! ˆ "<br />
!" = 0<br />
! ˆ<br />
! ˆ "<br />
!# = $ x ˆ sin # + y ˆ cos# = ˆ #<br />
! ˆ "<br />
!z = 0 ! ˆ "<br />
Path increment<br />
!ˆ<br />
! ˆ "<br />
!" = $ x ˆ cos" $ y ˆ sin " = $ ˆ #<br />
!z = 0<br />
z<br />
!" = 0<br />
!ˆ z<br />
!# = 0<br />
!ˆ z<br />
!z = 0<br />
We will have many uses for the path increment d r ! expressed in cylindrical coordinates:<br />
d r ! = d ! ˆ ! + zˆ z ! d! + !dˆ ! + ˆ z dz + zdˆ z<br />
( ) = ˆ<br />
$<br />
= ˆ ! d! + ! " ˆ ! "ˆ<br />
& d! + !<br />
"! "# d# + " ˆ !<br />
%<br />
"z dz ' $<br />
) + z ˆ dz + z &<br />
"ˆ z<br />
( %"! = ˆ ! d! + ˆ #!d# + z ˆ dz<br />
"ˆ<br />
d! + z<br />
"#<br />
"ˆ<br />
d# + z<br />
"z dz '<br />
)<br />
(
Time derivatives of the unit vectors<br />
We will also have many uses for the time derivatives of the unit vectors expressed in cylindrical coordinates:<br />
ˆ ˙ ! = "ˆ !<br />
"! ˙ ! + " ˆ ! ˙ # + " ˆ !<br />
"# "z z ˙ = ˆ # ˙ #<br />
ˆ ˙<br />
# = " ˆ #<br />
"! ˙ ! + " ˆ #<br />
#<br />
"# ˙ + " ˆ #<br />
z<br />
"z ˙ = $ ˆ ! ˙ #<br />
ˆ ˙ z = "ˆ z<br />
"! ˙ ! + "ˆ z ˙ # + "ˆ z<br />
"# "z z ˙ = 0<br />
Velocity and Acceleration<br />
The velocity and acceleration of a particle may be expressed in cylindrical coordinates by taking into account the associated<br />
rates of change in the unit vectors:<br />
!<br />
v = r ! ˙ = ˆ ˙<br />
!! + ˆ ! ˙ ! + ˆ ˙ z z + z ˆ z ˙ = ˆ ! ˙ ! + " ˆ ! ˙ " + z ˆ z ˙<br />
!<br />
v = ˆ ! ˙ ! + ˆ "! ˙ " + z ˆ z ˙<br />
!<br />
a = v ! ˙<br />
= ˆ ˙<br />
! ˙ ! + ˆ ! ˙ ! + ˆ ˙<br />
"! ˙ " + ˆ " ˙ ! ˙ " + ˆ "! ˙ " + z ˆ ˙<br />
z ˙ + z ˆ ˙ z<br />
= ˆ " ˙ " ˙ ! + ˆ ! ˙ ! # ˆ !! ˙ " 2 + " ˆ ˙ ! ˙ " + ˆ "! ˙ " + ˆ z ˙ z<br />
!<br />
a =<br />
( # 2<br />
) + ˆ<br />
ˆ ! ˙ ! " ! ˙<br />
( ) + ˆ<br />
# ! ˙ # + 2 ˙ ! ˙ #<br />
z ˙ z<br />
The del operator from the definition of the gradient<br />
Any (static) scalar field u may be considered to be a function of the cylindrical coordinates ! , ! , and z. The value of u<br />
changes by an infinitesimal amount du when the point of observation is changed by d r ! . That change may be determined<br />
from the partial derivatives as<br />
du = !u !u !u<br />
d" + d# +<br />
!" !# !z dz .<br />
But we also define the gradient in such a way as to obtain the result<br />
!<br />
du = ! u " d ! r<br />
Therefore,<br />
!u !u !u<br />
d" + d# +<br />
!" !# !z dz = !<br />
$ u% d r<br />
!<br />
or, in cylindrical coordinates,<br />
!u !u !u<br />
d" + d# +<br />
!" !# !z dz = $ !<br />
u<br />
( ) d" + $ !<br />
u<br />
"<br />
( ) "d# + $ !<br />
u<br />
#<br />
( ) dz<br />
z<br />
and we demand that this hold for any choice of d! , d! and dz. Thus,<br />
!<br />
(! u) = #u<br />
"<br />
from which we find<br />
#" , !<br />
! u<br />
( ) $<br />
= 1 "<br />
!<br />
! = ˆ<br />
#<br />
"<br />
#" + ˆ $ #<br />
" #$ + ˆ z # #z<br />
#u<br />
#$ , !<br />
! u<br />
( ) z<br />
= #u<br />
#z ,
Divergence<br />
!<br />
The divergence ! " A !<br />
is carried out taking into account, once again, that the unit vectors themselves are functions of the<br />
coordinates. Thus, we have<br />
!<br />
! " !<br />
A =<br />
&<br />
ˆ<br />
$<br />
#<br />
$# + % ˆ $<br />
# $% + z ˆ $ )<br />
(<br />
+ " A<br />
'<br />
$z # ˆ # + A ˆ<br />
%<br />
*<br />
% + A z ˆ z<br />
( )<br />
where the derivatives must be taken before the dot product so that<br />
!<br />
! " A ! &<br />
= ˆ<br />
$<br />
#<br />
$# + % ˆ $<br />
# $% + z ˆ $ )<br />
(<br />
+ " A<br />
!<br />
'<br />
$z*<br />
= ˆ # " $ A<br />
!<br />
$# + ˆ %<br />
# " $! A<br />
$% + z ˆ " $ A<br />
!<br />
$z<br />
&<br />
= ˆ # "(<br />
'<br />
$A #<br />
$# ˆ # + $A %<br />
$#<br />
ˆ % + $A z<br />
$# ˆ z + A $ ˆ #<br />
#<br />
$# + A $ ˆ %<br />
%<br />
$# + A $ˆ z )<br />
z +<br />
$# *<br />
+ ˆ %<br />
# " & $A #<br />
$% ˆ # + $A %<br />
$% ˆ % + $A z<br />
$% z ˆ + A $ˆ #<br />
#<br />
$% + A $ ˆ %<br />
%<br />
$% + A $ˆ z )<br />
(<br />
z<br />
'<br />
$%<br />
+<br />
*<br />
&<br />
+ z ˆ " (<br />
'<br />
$A #<br />
$z<br />
ˆ # + $A %<br />
$z<br />
ˆ % + $A z<br />
$z ˆ z + A #<br />
$ ˆ #<br />
$z + A %<br />
$ ˆ %<br />
$z + A z<br />
With the help of the partial derivatives previously obtained, we find<br />
!<br />
! " A !<br />
=<br />
& $A<br />
ˆ # " #<br />
$# ˆ # + $A % ˆ % + $A z<br />
(<br />
' $# $# ˆ z + 0 + 0 + 0 )<br />
+<br />
*<br />
ˆ %<br />
+<br />
# " & $A #<br />
$% ˆ # + $A % ˆ % + $A z<br />
z<br />
$% $% ˆ + A # % ˆ<br />
)<br />
(<br />
, A % ˆ # + 0+<br />
'<br />
*<br />
&<br />
+ z ˆ "(<br />
'<br />
$A #<br />
$z<br />
ˆ # + $A %<br />
$z<br />
ˆ % + $A z<br />
z<br />
$z ˆ + 0 + 0 + 0<br />
&<br />
= $A # ) &<br />
( + + 1 $A %<br />
' $# * # $% + A # ) &<br />
(<br />
+ + $A z )<br />
( +<br />
' # * ' $z *<br />
&<br />
= $A #<br />
$# + A # )<br />
(<br />
' #<br />
+ + 1 $A %<br />
* # $% + $A z<br />
$z<br />
!<br />
! " A !<br />
= 1 $<br />
# $# A ##<br />
( ) + 1 #<br />
$A %<br />
$% + $A z<br />
$z<br />
)<br />
+<br />
*<br />
$ˆ z )<br />
+<br />
$z*
Curl<br />
!<br />
The curl ! " A !<br />
is also carried out taking into account that the unit vectors themselves are functions of the coordinates.<br />
Thus, we have<br />
!<br />
! " A ! &<br />
= ˆ<br />
$<br />
#<br />
$# + ˆ % $<br />
# $% + ˆ z $ )<br />
(<br />
+ " A<br />
'<br />
$z # ˆ # + A ˆ<br />
%<br />
*<br />
% + A z ˆ z<br />
( )<br />
where the derivatives must be taken before the cross product so that<br />
!<br />
! " A ! &<br />
= ˆ<br />
$<br />
#<br />
$# + ˆ % $<br />
# $% + ˆ z $ )<br />
(<br />
+ " A<br />
!<br />
'<br />
$z*<br />
= ˆ # " $! A<br />
$# + ˆ %<br />
# " $! A<br />
$% + z ˆ " $! A<br />
$z<br />
&<br />
= ˆ # " $A #<br />
$# ˆ # + $A %<br />
% ˆ + $A z<br />
(<br />
$# $# z ˆ + A #<br />
'<br />
+ ˆ %<br />
# " & $A #<br />
$% ˆ # + $A %<br />
$% ˆ % + $A z<br />
(<br />
$% ˆ<br />
'<br />
&<br />
+ ˆ z " $A #<br />
(<br />
' $z<br />
ˆ # + $A %<br />
$z<br />
ˆ % + $A z<br />
$z ˆ z + A #<br />
$ˆ #<br />
$# + A %<br />
z + A $ˆ #<br />
#<br />
$% + A %<br />
$ ˆ #<br />
$z + A %<br />
$ ˆ %<br />
$# + A z<br />
$ˆ z )<br />
+<br />
$# *<br />
$ ˆ %<br />
$% + A z<br />
$ ˆ %<br />
$z + A z<br />
With the help of the partial derivatives previously obtained, we find<br />
!<br />
! " A ! &<br />
= ˆ # " $A #<br />
$# ˆ # + $A %<br />
% ˆ + $A z<br />
(<br />
' $# $# z ˆ + 0 + 0 + 0 )<br />
+<br />
*<br />
+ ˆ %<br />
# " & $A #<br />
(<br />
' $% ˆ<br />
&<br />
+ ˆ z " $A #<br />
(<br />
' $z<br />
# + $A %<br />
$%<br />
ˆ # + $A %<br />
$z<br />
ˆ % + $A z<br />
$% z ˆ + A ˆ<br />
#% , A % ˆ<br />
ˆ % + $A z<br />
$z ˆ z + 0 + 0 + 0<br />
)<br />
+<br />
*<br />
)<br />
# + 0+<br />
*<br />
&<br />
= $A %<br />
$# ˆ z , $A z ˆ ) &<br />
(<br />
% + + , 1 $A #<br />
' $# * # $% z ˆ + 1 $A z<br />
# $% ˆ # + A %<br />
(<br />
'<br />
# z ˆ )<br />
+<br />
*<br />
&<br />
+ $A #<br />
ˆ % , $A %<br />
(<br />
' $z $z ˆ # )<br />
+<br />
*<br />
$ˆ z )<br />
$%<br />
+<br />
*<br />
$ˆ z )<br />
+<br />
$z*<br />
& 1 $A<br />
= ˆ # z<br />
# $% , $A % )<br />
(<br />
+ + ˆ & $A #<br />
%<br />
' $z * $z , $A ) &<br />
z<br />
( + + z ˆ<br />
$A %<br />
' $# * $# + A %<br />
# , 1 $A # )<br />
(<br />
+<br />
'<br />
# $% *<br />
!<br />
! " A ! '<br />
= ˆ<br />
1 $A<br />
# z<br />
# $% & $A % *<br />
)<br />
( $z<br />
, + ˆ ' $A #<br />
%<br />
+ $z & $A * '<br />
z<br />
)<br />
( $#<br />
, + z ˆ<br />
1 $<br />
)<br />
+ # $# A % #<br />
(<br />
( ) & 1 #<br />
$A # *<br />
$%<br />
,<br />
+
Laplacian<br />
The Laplacian is a scalar operator that can be determined from its definition as<br />
!<br />
! 2 u = ! "<br />
!<br />
! u<br />
&<br />
( ) = (<br />
ˆ<br />
$<br />
#<br />
$# + % ˆ $<br />
# $% + z ˆ<br />
$ ) &<br />
'<br />
$z<br />
+ " ˆ # $u<br />
* $# + % ˆ $u<br />
# $% + z ˆ<br />
$u )<br />
(<br />
'<br />
$z<br />
+<br />
*<br />
= ˆ # " $ &<br />
$# ˆ # $u<br />
$# + ˆ % $u<br />
# $% + z ˆ<br />
$u )<br />
(<br />
+<br />
'<br />
$z *<br />
%<br />
+ ˆ<br />
# " $ &<br />
$% ˆ # $u<br />
$# + ˆ % $u<br />
# $% + z ˆ<br />
$u )<br />
(<br />
+<br />
'<br />
$z *<br />
+ z ˆ " $ $z<br />
&<br />
ˆ # $u<br />
$# + ˆ % $u<br />
# $% + z ˆ $u )<br />
(<br />
+<br />
'<br />
$z*<br />
With the help of the partial derivatives previously obtained, we find<br />
'<br />
! 2 u = ˆ " # ˆ " $ 2 u<br />
$" 2 % ˆ & $u<br />
" 2 $& +<br />
ˆ & $ 2 u<br />
" $&$" + z ˆ<br />
$ 2 u *<br />
)<br />
,<br />
(<br />
$z$" +<br />
&<br />
+ ˆ<br />
" # '<br />
ˆ $u<br />
&<br />
$" + ˆ<br />
$ 2 u<br />
"<br />
$"$& % ˆ " $u<br />
" $& +<br />
ˆ & $ 2 u<br />
" $& 2 + z ˆ $ 2 u *<br />
)<br />
,<br />
(<br />
$z$& +<br />
'<br />
+ z ˆ # ˆ<br />
$ 2 u ˆ<br />
"<br />
$"$z + & $ 2 u<br />
" $&$z + z ˆ<br />
$ 2 u*<br />
)<br />
(<br />
$z 2 ,<br />
+<br />
= $ 2 u<br />
$" 2 + 1 $u<br />
" $" + 1 $ 2 u<br />
" 2 $& 2 + $ 2 u<br />
$z 2<br />
= 1 $ '<br />
" $" " $u *<br />
) , + 1 $ 2 u<br />
( $" + " 2 $& 2 + $ 2 u<br />
$z 2<br />
Thus, the Laplacian operator can be written as<br />
! 2 = 1 # $<br />
" #" " # '<br />
& ) + 1 # 2<br />
% #" ( " 2 #* 2 + # 2<br />
#z 2