z
Spherical Coordinates
^
r
Transforms
! r
The forward and reverse coordinate transformations are
r=
x2 + y2 + z 2
y = r sin! sin"
z = r cos !
& = arctan ( y, x )
^
!
r
x = r sin ! cos"
! = arctan "# x 2 + y 2 , z$%
^
"
y
x
"
where we formally take advantage of the two argument arctan
function to eliminate quadrant confusion.
Unit Vectors
The unit vectors in the spherical coordinate system are functions of position. It is convenient to express them in terms of
the spherical coordinates and the unit vectors of the rectangular coordinate system which are not themselves functions of
position.
!
ˆ
ˆ ˆ
ˆr = r = xx + yy + zz = xˆ sin! cos " + yˆ sin! sin " + zˆ cos!
r
r
ˆ
ˆ
z
#
r
"ˆ =
= $ xˆ sin " + yˆ cos "
sin!
!ˆ = "ˆ # rˆ = xˆ cos ! cos" + yˆ cos! sin " $ zˆ sin!
Variations of unit vectors with the coordinates
Using the expressions obtained above it is easy to derive the following handy relationships:
!rˆ
=0
!r
!rˆ
= xˆ cos" cos # + yˆ cos " sin # $ ˆz sin " = "ˆ
!"
!rˆ
= $ xˆ sin " sin # + ˆy sin " cos # = ($ xˆ sin # + ˆy cos # ) sin" = #ˆ sin"
!#
!"ˆ
=0
!r
!"ˆ
=0
!#
!"ˆ
= $ xˆ cos " $ yˆ sin " = $ rˆ sin # + #ˆ cos #
!"
(
)
!"ˆ
=0
!r
!"ˆ
= # xˆ sin " cos $ # yˆ sin" sin $ # zˆ cos " = # rˆ
!"
!"ˆ
= # xˆ cos" sin $ + yˆ cos" cos $ = $ˆ cos"
!$
Path increment
!
We will have many uses for the path increment dr expressed in spherical coordinates:
$ !rˆ
!
!rˆ
!rˆ '
dr = d (rrˆ ) = rˆdr + rdrˆ = rˆdr + r& dr +
d" +
d# )
!"
!# (
% !r
= rˆdr + "ˆrd" + #ˆr sin "d#
Time derivatives of the unit vectors
We will also have many uses for the time derivatives of the unit vectors expressed in spherical coordinates:
!rˆ
!rˆ ˙ !rˆ ˙ ˆ ˙ ˆ ˙
r˙ˆ =
r˙ +
"+
# = " " + ## sin "
!r
!"
!#
˙ !"ˆ
!"ˆ ˙ !"ˆ ˙
"ˆ =
r˙ +
"+
# = $ rˆ"˙ + #ˆ #˙ cos "
!r
!"
!#
˙ !#ˆ
!#ˆ ˙ !#ˆ ˙
#ˆ =
r˙ +
"+
# = $ rˆ sin " + "ˆ cos " #˙
!r
!"
!#
(
)
Velocity and Acceleration
The velocity and acceleration of a particle may be expressed in spherical coordinates by taking into account the associated
rates of change in the unit vectors:
! !
v = r˙ = r˙ˆr + rˆr˙
!
v = rˆr˙ + !ˆr!˙ + "ˆr"˙ sin!
! !
˙
˙
a = v˙ = r˙ˆr˙ + rˆ r˙˙ + !ˆr!˙ + !ˆr˙!˙ + !ˆr!˙˙ + "ˆr"˙ sin! + "ˆr˙"˙ sin! + "ˆr"˙˙sin ! + "ˆr"˙!˙ cos !
(
)
(
)
= !ˆ!˙ + "ˆ"˙ sin ! r˙ + rˆr˙˙ + # rˆ !˙ + "ˆ"˙ cos! r!˙ + !ˆr˙!˙ + !ˆr!˙˙
[(
)]
+ # rˆ sin! + !ˆ cos! "˙ r"˙sin ! + "ˆr˙"˙ sin ! + "ˆr"˙˙sin ! + "ˆr"˙!˙ cos !
(
) (
) (
!
a = r̂ ""
r ! r"" 2 ! r#" 2 sin 2 " + "ˆ r""" + 2 r""" ! r#" 2 sin " cos " + #̂ r#"" sin " + 2r""#" cos " + 2 r"#" sin "
)
The del operator from the definition of the gradient
Any (static) scalar field u may be considered to be a function of the spherical coordinates r, θ, and φ. The value of u
!
changes by an infinitesimal amount du when the point of observation is changed by dr . That change may be determined
from the partial derivatives as
du =
!u
!u
!u
dr +
d" +
d# .
!r
!"
!#
But we also define the gradient in such a way as to obtain the result
!
!
du = !u " dr
Therefore,
!
!u
!u
!u
!
dr +
d" +
d# = $u % dr
!r
!"
!#
or, in spherical coordinates,
!
!
!
!u
!u
!u
dr +
d" +
d# = $u dr + $u rd" + $ u r sin" d#
r
"
#
!r
!"
!#
( )
( )
( )
and we demand that this hold for any choice of dr, dθ, and d φ. Thus,
!
(!u )
r
=
"u
,
"r
!
(!u )
#
=
1 "u
,
r "#
!
(!u )
$
=
1 "u
,
r sin # "$
from which we find
!
" #ˆ "
$ˆ "
! = rˆ +
+
"r r "# r sin # "$
Divergence
! !
The divergence ! " A is carried out taking into account, once again, that the unit vectors themselves are functions of the
coordinates. Thus, we have
! ! & # $ˆ #
%ˆ # )
! " A = ( rˆ +
+
+ " Ar rˆ + A$ $ˆ + A% %ˆ
' #r r #$ r sin $ #% *
(
)
where the derivatives must be taken before the dot product so that
! ! & # $ˆ #
%ˆ # ) !
! " A = ( rˆ +
+
+ "A
' #r r #$ r sin $ #% *
!
!
!
ˆ #A
ˆ
#
A
$
%
#
A
= rˆ "
+ "
+
"
#r r #$ r sin$ #%
& #A
#A% ˆ
#A
#rˆ
#$ˆ
#%ˆ )
= rˆ " ( r rˆ + $ $ˆ +
% + Ar
+ A$
+ A%
+
#r
#r
#r
#r
#r *
' #r
#A% ˆ
$ˆ & #A
#A
#rˆ
#$ˆ
#%ˆ )
+ " ( r rˆ + $ $ˆ +
% + Ar
+ A$
+ A%
+
r ' #$
#$
#$
#$
#$
#$ *
+
& #A
#A% ˆ
%ˆ
#A
#rˆ
#$ˆ
#%ˆ )
" ( r rˆ + $ $ˆ +
% + Ar
+ A$
+ A%
+
r sin$ ' #%
#%
#%
#%
#%
#% *
With the help of the partial derivatives previously obtained, we find
! !
& #A
#A
)
#A
! " A = rˆ " ( r rˆ + $ $ˆ + % %ˆ + 0 + 0 + 0+
#r
#r
' #r
*
ˆ
#A
#A
)
$ & #A
+ " ( r rˆ + $ $ˆ + % %ˆ + Ar$ˆ + A$ (, rˆ ) + 0+
r ' #$
#$
#$
*
+
%ˆ
r sin$
[(
& #A ) & 1 #A$ Ar ) & 1 #A% Ar A$ cos$ )
= ( r + +(
+ + +(
+
+
+
' #r * ' r #$
r * ' r sin$ #%
r
r sin $ *
2A ) & 1 #A$ A$ cos$ )
1 #A%
& #A
=( r + r+ +(
+
++
' #r
r * ' r #$
r sin $ * r sin $ #%
! ! 1 # 2
#A%
1 #
( A$ sin$ ) + 1
!" A = 2
r Ar +
r #r
r sin$ #$
r sin $ #%
(
)]
& #A
#A% ˆ
)
#A
" ( r rˆ + $ $ˆ +
% + Ar sin$ %ˆ + A$ cos $%ˆ + A% , rˆ sin $ + $ˆ cos$ +
#%
#%
' #%
*
)
Curl
! !
The curl ! " A is also carried out taking into account that the unit vectors themselves are functions of the coordinates.
Thus, we have
! ! & # $ˆ #
%ˆ # )
! " A = ( rˆ +
+
+ " Arrˆ + A$ $ˆ + A% %ˆ
' #r r #$ r sin $ #% *
(
)
where the derivatives must be taken before the cross product so that
! ! & # $ˆ #
%ˆ # ) !
! " A = ( rˆ +
+
+" A
' #r r #$ r sin $ #% *
!
!
!
ˆ #A
ˆ
#
A
$
%
#
A
= rˆ "
+ "
+
"
#r r #$ r sin $ #%
& #A
#A% ˆ
#A
#rˆ
#$ˆ
#%ˆ )
= rˆ " ( r rˆ + $ $ˆ +
% + Ar
+ A$
+ A% +
#r
#r
#r
#r
#r *
' #r
#A% ˆ
$ˆ & #A
#A
#rˆ
#$ˆ
#%ˆ )
+ " ( r rˆ + $ $ˆ +
% + Ar
+ A$
+ A%
+
r ' #$
#$
#$
#$
#$
#$ *
+
& #A
#A% ˆ
%ˆ
#A
#rˆ
#$ˆ
#%ˆ )
" ( r rˆ + $ $ˆ +
% + Ar
+ A$
+ A%
+
r sin$ ' #%
#%
#%
#%
#%
#% *
With the help of the partial derivatives previously obtained, we find
! !
& #A
#A% ˆ
)
#A
! " A = rˆ " ( r rˆ + $ $ˆ +
% + 0 + 0 + 0+
#r
#r
' #r
*
#A
)
$ˆ & #A
#A
+ " ( r rˆ + $ $ˆ + % %ˆ + Ar$ˆ + A$ ( ,ˆr ) + 0+
r ' #$
#$
#$
*
ˆ
#A
& #A
#A
)
%
+
" ( r rˆ + $ $ˆ + % %ˆ + Ar sin $%ˆ + A$ cos $%ˆ + A% , rˆ sin $ + $ˆ cos$ +
r sin$ ' #%
#%
#%
*
[(
& #A
#A
= ( $ %ˆ , %
#r
' #r
) & 1 #Ar ˆ 1 #A%
A )
$ˆ+ + ( ,
%+
rˆ + $ %ˆ+
r #$
r *
* ' r #$
A% ˆ A% cos$
& 1 #Ar ˆ
1 #A$
+(
$,
rˆ ,
$+
r sin$ #%
r
r sin$
' r sin$ #%
)
rˆ+
*
& 1 #A%
1 #A$ A% cos$ )
= rˆ (
,
+
+
r sin $ *
' r #$ r sin$ #%
& #A
1 #Ar A% )
+ $ˆ ( , % +
,
r sin $ #%
r +*
' #r
1 #Ar A$ )
& #A
+ %ˆ ( $ ,
+
+
' #r
r #$
r *
! !
!" A =
* %ˆ ' $
rˆ ' $
$A *
#ˆ '$Ar
$
$A *
A% sin # & # , +
& sin #
rA% , + ) ( rA# ) & r ,
)
)
r sin # ( $#
$% + r sin # ( $%
$r
$# +
+ r ( $r
(
)
( )
)]
Laplacian
The Laplacian is a scalar operator that can be determined from its definition as
! !
& # $ˆ #
%ˆ # ) & ˆ #u $ˆ #u
%ˆ #u )
! 2 u = ! " !u = ( rˆ +
+
+
+
+ "(r
+
' #r r #$ r sin $ #% * ' #r r #$ r sin$ #% *
( )
= rˆ "
# & #u $ˆ #u
%ˆ #u )
+
+
(rˆ
+
#r ' #r r #$ r sin $ #% *
$ˆ # & ˆ #u $ˆ #u
%ˆ #u )
"
+
+
(r
+
r #$ ' #r r #$ r sin $ #% *
%ˆ
# & ˆ #u $ˆ #u
%ˆ #u )
+
"
+
+
(r
+
r sin $ #% ' #r r #$ r sin$ #% *
+
With the help of the partial derivatives previously obtained, we find
' # 2 u %ˆ #u %ˆ # 2 u
&ˆ
#u
&ˆ # 2 u *
! 2 u = rˆ " ) rˆ 2 $ 2
+
$ 2
+
,
r #% r #%#r r sin % #& r sin% #&#r +
( #r
+
%ˆ ' ˆ #u
# 2u rˆ #u %ˆ # 2 u &ˆ cos % #u
&ˆ
# 2u *
" )%
+ rˆ
$
+
$
+
,
r ( #r
#r#% r #% r #% 2 r sin 2 % #& r sin % #&#% +
+
'
&ˆ
#u ˆ # 2 u &ˆ cos % #u %ˆ # 2 u rˆ sin % + %ˆ cos % #u
&ˆ # 2u *
" )&ˆ sin %
+r
+
+
$
+
,
r sin % (
#r
#r#&
r
#% r #%#&
r sin %
#& r sin % #& 2 +
' # 2 u * ' 1 #u 1 # 2 u * ' 1 #u
cos% #u
1
# 2u *
=) 2,+)
+ 2
+ 2
+ 2 2
2, +)
2,
( #r + ( r #r r #% + ( r #r r sin % #% r sin % #& +
' # 2 u 2 #u * ' 1 # 2u
cos% #u * '
1
# 2u *
=) 2 +
+
+
+
, )
, )
,
r #r + ( r 2 #% 2 r2 sin % #% + ( r 2 sin 2 % #& 2 +
( #r
=
1 # ' 2 #u *
1
# '
#u *
1
# 2u
)r
,+ 2
)sin % , + 2
2
2
r #r ( #r + r sin% #% (
#% + r sin % #& 2
Thus, the Laplacian operator can be written as
!2 =
1 " # 2 " &
1
" #
"&
1
"2
%r
(+ 2
%sin ) ( + 2 2
2
r "r $ "r ' r sin) ") $
") ' r sin ) "* 2