~Grad; Div; Rot; Lalpaciano de coordenadas ESFERICAS.

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Spherical Coordinates 
Transforms 
The forward and reverse coordinate transformations are 
 
r = x2 + y2 + z 2
! = arctan x2 + y2 , z" # 
$ 
% 
& = arctan y,x( )
 
x = rsin! cos"
y = r sin! sin"
z = rcos!
 
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 r =
! r 
r =
xˆ x + yˆ y + z ˆ z 
r =
ˆ x sin! cos" + ˆ y sin! sin" + ˆ z cos!
ˆ " = ˆ z # ˆ r sin! = $ ˆ x sin" + ˆ y cos"
ˆ ! = ˆ " # ˆ r = ˆ x cos! cos" + ˆ y cos! sin" $ ˆ z sin!
 
Variations of unit vectors with the coordinates 
Using the expressions obtained above it is easy to derive the following handy relationships: 
 
! ˆ r 
!r = 0
! ˆ r 
!"
= ˆ x cos" cos# + ˆ y cos" sin# $ ˆ z sin" = ˆ " 
! ˆ r 
!#
= $ ˆ x sin" sin# + ˆ y sin" cos# = $ ˆ x sin# + ˆ y cos #( )sin" = ˆ # sin"
 
 
! ˆ " 
!r = 0
! ˆ " 
!#
= 0
! ˆ " 
!"
= $ ˆ x cos" $ ˆ y sin" = $ ˆ r sin# + ˆ # cos#( )
 
 
! ˆ " 
!r = 0
! ˆ " 
!"
= # ˆ x sin" cos$ # ˆ y sin" sin$ # ˆ z cos" = # ˆ r 
! ˆ " 
!$
= # ˆ x cos" sin$ + ˆ y cos" cos$ = ˆ $ cos"
 
x
y
z
r^
!^
"^
!
"
r
r
Path increment 
We will have many uses for the path increment d
! r expressed in spherical coordinates: 
 
 
d! r = d r ˆ r ( ) = ˆ r dr + rd ˆ r = ˆ r dr + r ! ˆ r 
!r dr +
! ˆ r 
!"
d" + ! ˆ r 
!#
d#$ 
% 
& 
' 
( 
) 
= ˆ r dr + ˆ " rd" + ˆ # rsin"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: 
 
ˆ ˙ r = ! ˆ r 
!r ˙ r +
!ˆ r 
!"
˙ " + ! ˆ r 
!#
˙ # = ˆ " ˙ " + ˆ # ˙ # sin"
ˆ ˙ " = !
ˆ " 
!r ˙ r +
! ˆ " 
!"
˙ " + !
ˆ " 
!#
˙ # = $ ˆ r ˙ " + ˆ # ˙ # cos"
ˆ ˙ # = !
ˆ # 
!r ˙ r +
! ˆ # 
!"
˙ " + !
ˆ # 
!#
˙ # = $ ˆ r sin" + ˆ " cos"( )˙ # 
 
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 = ˆ ˙ r r + ˆ r ˙ r 
 
! v = ˆ r ˙ r + ˆ ! r ˙ ! + ˆ " r ˙ " sin! 
 
 
! a = ! ˙ v = ˆ ˙ r ˙ r + ˆ r ˙ ˙ r + ˆ ˙ ! r ˙ ! + ˆ ! ˙ r ˙ ! + ˆ ! r ˙ ˙ ! + ˆ ˙ " r ˙ " sin! + ˆ " ˙ r ˙ " sin! + ˆ " r ˙ ˙ " sin ! + ˆ " r ˙ " ˙ ! cos!
= ˆ ! ˙ ! + ˆ " ˙ " sin!( ) ˙ r + ˆ r ˙ ˙ r + # ˆ r ˙ ! + ˆ " ˙ " cos!( )r ˙ ! + ˆ ! ˙ r ˙ ! + ˆ ! r ˙ ˙ ! 
+ # ˆ r sin! + ˆ ! cos!( ) ˙ " [ ]r ˙ " sin ! + ˆ " ˙ r ˙ " sin! + ˆ " r ˙ ˙ " sin ! + ˆ " r ˙ " ˙ ! cos!
 
 
!a = rˆ ""r ! r "" 2 ! r "# 2 sin2 "( ) + "ˆ r""" + 2 "r "" ! r "# 2 sin" cos"( ) + #ˆ 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 d
! r . That change may be determined 
from the partial derivatives as 
 du = !u
!r dr +
!u
!"
d" + !u
!#
d# . 
But we also define the gradient in such a way as to obtain the result 
 du =
! 
! u " d! r 
Therefore, 
 
 
!u
!r dr +
!u
!"
d" + !u
!#
d# =
! 
$ u % d! r 
or, in spherical coordinates, 
 
 
!u
!r dr +
!u
!"
d" + !u
!#
d# =
! 
$ u( )r dr +
! 
$ u( )" rd" +
! 
$ u( )# r sin"d# 
and we demand that this hold for any choice of dr, dθ, and dφ. Thus, 
 
 
!
!u( )r =
"u
"r ,
!
!u( )# =
1
r
"u
"#
,
!
!u( )$ =
1
r sin#
"u
"$
, 
from which we find 
 
 
! 
! = ˆ r "
"r +
ˆ # 
r
"
"#
+
ˆ $ 
rsin#
"
"$ 
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 
 
 
! 
! "
! 
A = ˆ r #
#r +
ˆ $ 
r
#
#$
+
ˆ % 
rsin$
#
#%
& 
' 
( 
) 
* 
+ " Ar ˆ r + A$ ˆ $ + A% ˆ % ( ) 
where the derivatives must be taken before the dot product so that 
 
 
! 
! "
! 
A = ˆ r #
#r +
ˆ $ 
r
#
#$
+
ˆ % 
rsin$
#
#%
& 
' 
( 
) 
* 
+ "
! 
A 
= ˆ r " #
! 
A 
#r +
ˆ $ 
r "
#
! 
A 
#$
+
ˆ % 
rsin$ "
#
! 
A 
#%
= ˆ r " #Ar
#r ˆ r +
#A$
#r
ˆ $ + #A%
#r
ˆ % + Ar
# ˆ r 
#r + A$
# ˆ $ 
#r + A%
# ˆ % 
#r
& 
' 
( 
) 
* 
+ 
+
ˆ $ 
r "
#Ar
#$
ˆ r + #A$
#$
ˆ $ + #A%
#$
ˆ % + Ar
# ˆ r 
#$
+ A$
# ˆ $ 
#$
+ A%
# ˆ % 
#$
& 
' 
( 
) 
* 
+ 
+
ˆ % 
r sin$ "
#Ar
#%
ˆ r + #A$
#%
ˆ $ + #A%
#%
ˆ % + Ar
#ˆ r 
#%
+ A$
# ˆ $ 
#%
+ A%
# ˆ % 
#%
& 
' 
( 
) 
* 
+ 
 
With the help of the partial derivatives previously obtained, we find 
 
 
! 
! "
! A = ˆ r " #Ar
#r ˆ r +
#A$
#r
ˆ $ + #A%
#r
ˆ % + 0 + 0 + 0& 
' 
( 
) 
* 
+ 
+
ˆ $ 
r "
#Ar
#$
ˆ r + #A$
#$
ˆ $ + #A%
#$
ˆ % + Ar ˆ $ + A$ , ˆ r ( ) + 0
& 
' 
( 
) 
* 
+ 
+
ˆ % 
r sin$ "
#Ar
#%
ˆ r + #A$
#%
ˆ $ + #A%
#%
ˆ % + Ar sin$ ˆ % + A$ cos$ ˆ % + A% , ˆ r sin$ + ˆ $ cos$( )[ ]& ' ( 
) 
* 
+ 
= #Ar
#r
& 
' 
( 
) 
* 
+ +
1
r
#A$
#$
+ Arr
& 
' 
( 
) 
* 
+ +
1
r sin$
#A%
#%
+ Arr +
A$ cos$
rsin$
& 
' 
( 
) 
* 
+ 
=
#Ar
#r +
2Ar
r
& 
' 
( ) 
* 
+ +
1
r
#A$
#$
+
A$ cos$
r sin$
& 
' 
( ) 
* 
+ +
1
rsin$
#A%
#%
 
 
 
! 
! "
! 
A = 1r 2
#
#r r
2 Ar( ) + 1rsin$
#
#$
A$ sin$( ) + 1r sin$
#A%
#% 
Curl 
The curl 
! 
! "
! 
A is also carried out taking into account that the unit vectors themselves are functions of the coordinates. 
Thus, we have 
 
 
! 
! "
! 
A = ˆ r #
#r +
ˆ $ 
r
#
#$
+
ˆ % 
rsin$
#
#%
& 
' 
( 
) 
* 
+ " Ar ˆ r + A$ ˆ $ + A% ˆ % ( ) 
where the derivatives must be taken before the cross product so that 
 
 
! 
! "
! 
A = ˆ r #
#r +
ˆ $ 
r
#
#$
+
ˆ % 
rsin$
#
#%
& 
' 
( 
) 
* 
+ "
! 
A 
= ˆ r " #
! 
A 
#r +
ˆ $ 
r "
#
! 
A 
#$
+
ˆ % 
rsin$ "
#
! 
A 
#%
= ˆ r " #Ar
#r ˆ r +
#A$
#r
ˆ $ + #A%
#r
ˆ % + Ar
#ˆ r 
#r + A$
# ˆ $ 
#r + A%
# ˆ % 
#r
& 
' 
( 
) 
* 
+ 
+
ˆ $ 
r "
#Ar
#$
ˆ r + #A$
#$
ˆ $ + #A%
#$
ˆ % + Ar
# ˆ r 
#$
+ A$
# ˆ $ 
#$
+ A%
# ˆ % 
#$
& 
' 
( 
) 
* 
+ 
+
ˆ % 
r sin$ "
#Ar
#%
ˆ r + #A$
#%
ˆ $ + #A%
#%
ˆ % + Ar
# ˆ r 
#%
+ A$
# ˆ $ 
#%
+ A%
# ˆ % 
#%
& 
' 
( 
) 
* 
+ 
 
With the help of the partial derivatives previously obtained, we find 
 
 
! 
! "
! 
A = ˆ r " #Ar
#r ˆ r +
#A$
#r
ˆ $ + #A%
#r
ˆ % + 0 + 0 + 0& 
' 
( 
) 
* 
+ 
+
ˆ $ 
r "
#Ar
#$
ˆ r + #A$
#$
ˆ $ + #A%
#$
ˆ % + Ar ˆ $ + A$ ,ˆ r ( ) + 0
& 
' 
( 
) 
* 
+ 
+
ˆ % 
r sin$ "
#Ar
#%
ˆ r + #A$
#%
ˆ $ + #A%
#%
ˆ % + Ar sin$ ˆ % + A$ cos$ ˆ % + A% , ˆ r sin$ + ˆ $ cos$( )[ ]& ' ( 
) 
* 
+ 
= #A$
#r
ˆ % , #A%
#r
ˆ $ & 
' 
( 
) 
* 
+ + ,
1
r
#Ar
#$
ˆ % + 1r
#A%
#$
ˆ r + A$r
ˆ % & 
' 
( 
) 
* 
+ 
+
1
r sin$
#Ar
#%
ˆ $ , 1r sin$
#A$
#%
ˆ r , A%r
ˆ $ + A% cos$r sin$ ˆ r 
& 
' 
( 
) 
* 
+ 
= ˆ r 1r
#A%
#$
, 1r sin$
#A$#%
+
A% cos$
rsin$
& 
' 
( 
) 
* 
+ 
+ ˆ $ , #A%
#r +
1
rsin$
#Ar
#%
,
A%
r
& 
' 
( 
) 
* 
+ 
+ ˆ % #A$
#r ,
1
r
#Ar
#$
+
A$
r
& 
' 
( ) 
* 
+ 
 
 
 
! 
! "
! 
A = ˆ r r sin#
$
$#
A% sin#( ) & $A#$%
' 
( ) 
* 
+ , 
+
ˆ # 
rsin#
$Ar
$%
& sin# $
$r rA%( )
' 
( ) 
* 
+ , 
+
ˆ % 
r
$
$r rA#( ) &
$Ar
$#
' 
( ) 
* 
+ , 
Laplacian 
The Laplacian is a scalar operator that can be determined from its definition as 
 
 
!2u =
! 
! "
! 
! u( ) = ˆ r ##r +
ˆ $ 
r
#
#$
+
ˆ % 
rsin$
#
#%
& 
' 
( 
) 
* 
+ " ˆ r #u#r +
ˆ $ 
r
#u
#$
+
ˆ % 
r sin$
#u
#%
& 
' 
( 
) 
* 
+ 
= ˆ r " #
#r ˆ r 
#u
#r +
ˆ $ 
r
#u
#$
+
ˆ % 
rsin$
#u
#%
& 
' 
( 
) 
* 
+ 
+
ˆ $ 
r "
#
#$
ˆ r #u
#r +
ˆ $ 
r
#u
#$
+
ˆ % 
rsin$
#u
#%
& 
' 
( 
) 
* 
+ 
+
ˆ % 
rsin$ "
#
#%
ˆ r #u
#r +
ˆ $ 
r
#u
#$
+
ˆ % 
r sin$
#u
#%
& 
' 
( 
) 
* 
+ 
 
With the help of the partial derivatives previously obtained, we find 
 
!2u = ˆ r " ˆ r #
2u
#r 2 $
ˆ % 
r 2
#u
#%
+
ˆ % 
r
# 2u
#%#r $
ˆ & 
r2 sin%
#u
#&
+
ˆ & 
r sin%
# 2u
#&#r
' 
( 
) 
* 
+ 
, 
+
ˆ % 
r "
ˆ % #u
#r + ˆ r 
# 2u
#r#% $
ˆ r 
r
#u
#%
+
ˆ % 
r
# 2u
#% 2
$
ˆ & cos%
r sin2%
#u
#&
+
ˆ & 
rsin%
# 2u
#&#%
' 
( 
) 
* 
+ 
, 
+
ˆ & 
rsin% "
ˆ & sin % #u
#r +
ˆ r #
2u
#r#& +
ˆ & cos%
r
#u
#%
+
ˆ % 
r
# 2u
#%#&
$
ˆ r sin% + ˆ % cos%
rsin%
#u
#&
+
ˆ & 
rsin%
# 2u
#&2
' 
( 
) 
* 
+ 
, 
= #
2u
#r 2
' 
( 
) 
* 
+ 
, +
1
r
#u
#r +
1
r2
# 2u
#% 2
' 
( 
) 
* 
+ 
, +
1
r
#u
#r +
cos%
r 2 sin%
#u
#%
+ 1r2 sin2 %
# 2u
#&2
' 
( 
) 
* 
+ 
, 
= #
2u
#r 2 +
2
r
#u
#r
' 
( 
) 
* 
+ 
, +
1
r 2
# 2u
#%2
+ cos%r2 sin%
#u
#%
' 
( 
) 
* 
+ 
, +
1
r 2 sin2%
# 2u
#&2
' 
( 
) 
* 
+ 
, 
= 1r 2
#
#r r
2 #u
#r
' 
( 
) 
* 
+ 
, +
1
r 2 sin%
#
#%
sin% #u
#%
' 
( 
) 
* 
+ 
, +
1
r2 sin 2%
# 2u
#&2
 
Thus, the Laplacian operator can be written as 
 !2 = 1r 2
"
"r r
2 "
"r
# 
$ 
% & 
' 
( +
1
r 2 sin)
"
")
sin) "
")
# 
$ 
% & 
' 
( +
1
r2 sin2 )
" 2
"*2